In the article "Morrey spaces, their duals and preduals", by Marcel Rosenthal and Hans Triebel, for every $1 \leqslant p < \infty$ and $-\frac{n}{p} < r < 0$ the Morrey Spaces are defined as
$$ L^r_p(\mathbb R^n) := \left\{f \in L^p_{\text{loc}} \, \colon \, \| f \|_{L^r_p(\mathbb R^n)} < \infty \right\}, $$
where
$$ \| f \|_{L_p^r(\mathbb R^n)} := \sup_{J \in \mathbb N_0, \, M \in \mathbb Z^n} 2^{J(\frac{n}{p}+r)} \| f \|_{L^p(Q_{J,M})}, $$
where $Q_{J,M}$ represents the usual dyadic cubes in $\mathbb R^n$ with sides of length $2^{-J}$ parallel to the axes of coordinates and $2^{-J}M$ as the lower left corner. At some point (Proposition $3.7$) we have the result
Propostion. Let $1 \leqslant p < \infty$ and $-n/p < r < 0$. Then the space $L^r_p(\mathbb R^n)$ is non-separable.
Proof. Let $$ Q_{J_l,M^l} = 2^{-J_l}M^l + 2^{-J_l}(0,1)^n, \quad l \in \mathbb N $$ be disjoint cubes with $Q_{J_l,M^l} \subset Q = (0,1)^n,$ where $J_l \in \mathbb N, J_1 < J_2 < \dots $ and $M^l \in \mathbb Z^n.$ Let $$ f^\lambda = \sum_{l=1}^\infty \lambda_{J_l,M^l}2^{-J_lr}\chi_{J_l,M^l}, $$ where $\chi_{J_l,M^l}$ is the charateristic function of $Q_{J_l,M^l}$ and $\lambda = \{\lambda_{J_l,M^l} \}_{l=1}^\infty$ with either $\lambda_{J_l,M^l} = 1$ or $\lambda_{J_l,M^l} = -1$. Let $J \in \mathbb Z$ and $M \in \mathbb Z^n$. Then $$ 2^{J(\frac{n}{p}+r)} \int_{Q_{J,M}} |f^\lambda(x)|^p \, dx \leqslant \sum_{l \colon J_l \geqslant J} 2^{-(J_l-J)(\frac{n}{p}+r)p} + \sum_{l \colon J_l < J} 2^{(J-J_l)rp} < \infty $$ where we used $\frac{n}{p}+r > 0$ and $r > 0$. Hence $f^\lambda \in L^r_p(\mathbb R^n)$. If $\lambda^1$ and $\lambda^2$ are two different admitted sequences then one has $\lambda^1_{J_l,M^l} = 1$ and $\lambda_{J_l,M^l} = -1$ for some $l \in \mathbb N$ and $$ \| f^{\lambda^1} - f^{\lambda^2} \|_{L^r_p(\mathbb R^n)} \geqslant 2^{J_l(\frac{n}{p}+r)} \left( \int_{Q_{J_l,M^l}} |(f^{\lambda^1} - f^{\lambda^2})(x)|^p \, dx \right)^{1/p} = 2. $$ But the set of all these admitted functions $f^\lambda$ is non-countable, having the cardinality of $\mathbb R$. Thus it follows that $L^r_p(\mathbb R^n)$ is non-separable. $ \blacksquare$
I find the proof quite hard to understand, perhaps because I don't have much experience in dealing with the problem of separability of function spaces. Either way, the main purpose of my question is not the proof itself. Most of the time, when reading texts from different authors, Morrey spaces won't be defined in the same way. I am most used to the following definition:
Consider $1 \leqslant p < \infty$ and $0 < \lambda < n$ arbitrarily. Then the Morrey Space $L^{p,\lambda}(\mathbb R^n)$ is defined as
$$ L^{p,\lambda} := \left\{ f \in L^p_{\text{loc}}(\mathbb R^n) \, \colon \sup_{x \in \mathbb R^n, \, r > 0} r^{-\lambda} \| f \|_{L^p(B(x,r)}^p < \infty \right\}. $$
Now, my main goal is to prove that the Morrey Spaces (defined as I just showed) are non-separable for every $1 \leqslant p < \infty$ and $0 < \lambda < n$. For this, I think that it might be possible to adapt the proof provided by Triebel and Rosenthal that I've quoted above, but I am having a hard time doing so.
I appreciate any help, hints or answers in advance.
First of all, I do want to comment on the usual way to show inseparability for a normed function space $(X, \|\cdot\|)$. So you need to find an uncountable subset $F\subset X$ in your space where any two points have a fixed distance from each other. That is, $\exists \epsilon_0>0$ such that $\forall f, f'\in F$, we have $\|f-f'\|\geq \epsilon_0$.
We now argue that any dense subset $Y\subset X$, that is, the closure $\bar Y=X$, must be uncountable. For consider the open balls $\{B(f, \epsilon_0/2)\,|\, f\in F\}$. By the fixed distance condition, these balls are disjoint. Now since $\bar Y=X$, $\exists y_f\in Y\cap B(f, \epsilon/2)$ by the characterization of closure. The $y_f$ are distinct (since the balls don't intersect), so $\{y_f\,|\,f\in F\}$ is uncountable. So is $Y$ by $\{y_f\,|\,f\in F\}\subset Y$.
The main way to find an uncountable set is by the power set $2^{\mathbb N}$, that is, the set of subsets of ${\mathbb N}$.
This happens for example for $\ell^{\infty}$. For any subset $I\subset {\mathbb N}$, define $$ (e_I)_i=\begin{cases} 1 & i\in I\\ 0 & i\notin I \end{cases}. $$ Then $\|e_I-e_{I'}\|_\infty = 1$ if $I\neq I'$. It also happens for your example of $L^r_p({\mathbb R}^n)$, where you pick any subset $I\subset {\mathbb N}$ and define $$ \lambda^I_{J_l, M^l}= \begin{cases} 1 & l\in I,\\ -1 & l\notin I. \end{cases} $$
Now we come to the other version of the Morrey space. The whole point is to establish a dictionary between the two spaces, so that we can translate the idea from one to the other. There is an unfortunate double use of the letters $r$ in the old and new spaces. I will have to settle to call the radius of a ball by $s$. This also happened for the $\pm 1$-valued function $\lambda$, and I will call it by $\kappa$ in the new version, and I will make it take on $1$ or $0$, since 2 is not so favored if we are not doing things dyadically.
So for $1\leq p<\infty$, and $0<\lambda <n$, define for $f\in L^p_{\text{loc}}({\mathbb R}^n)$, $$ \|f\|_{L^{p, \lambda}({\mathbb R}^n)} = \sup_{x\in {\mathbb R}^n, s>0} s^{-\frac{\lambda}p} \|f\|_{L^p(B(x, s))}. $$ (So we just take the $1/p$-th power in your definition to have homogeneity and get a norm.)
The idea is to draw the analogue between a dyadic cube $Q_{J, M}$ and a ball $B(x, s)$. So the lower left point $M\in {\mathbb Z}^n$ corresponds to the center $x\in {\mathbb R}^n$, and the side-length $2^{-J}$ corresponds to the radius $s$.
If you compare the two norms $\|\cdot\|_{L^{r}_p}$ and $\|\cdot\|_{L^{p, \lambda}}$, you see we should have the correspondence $$ 2^{J(\frac{n}{p} + r)} = s^{-\frac{\lambda}{p}}. $$ By $2^{-J}=s$, we get $$ \frac{n}{p}+r=\frac{\lambda}{p}. $$ This would totally match with the conditions of $-\frac{n}{p} < r < 0$ and $0<\lambda<n$.
Now let $\kappa:{\mathbb N}\to \{0, 1\}$. Let $s_l=2^{-l}\searrow 0$. Let $x_l\in {\mathbb R}^n$ be chosen such that the distance $d(x_l, x_{l'})=\|x_l-x_{l'}\|\geq 2$. Then the balls $B(x_l, s_l)$ are disjoint. (We do need the sequence of radius to go down to 0 in an exponential fashion, and this is automatic in the dyadic version where we have $2^{-J_l}$ with $J_1<J_2<\dots$. Also, we need to require the centers to be a fixed distance apart, which again is automatic in the dyadic version for the lattice points.)
Let $$ f^\kappa = \sum_{l=1}^\infty \kappa(l) s_l^{\frac{\lambda-n}{p}} \chi_l, $$ where $\chi_l = \chi_{B(x_l, s_l)}$ is the characteristic function of the ball $B(x_l, s_l)$. (Again, this is borrowed from the dyadic version using our dictionary.)
Now we show two things. First, $f^\kappa\in L^{p, \lambda}({\mathbb R}^n)$, and second, two different such functions have a fixed distance apart.
Consider a general ball $B(x, s)$. Let $\alpha_n = \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}$ denote the volume of the unit ball in ${\mathbb R}^n$. Note first that since the balls $B(x_l,s_l)$ are disjoint, the different summands don't interact with each other, and we have $$ |f^\kappa|^p = \sum_{l=1}^\infty |\kappa(l)|^p s_l^{\lambda-n}\chi_l.\tag{1} $$
We have to deal with two cases. First $s \geq\frac{1}{2}$, then $s\geq s_l=2^{-l}$ for $l\geq 1$. We have \begin{align*} s^{-\lambda}\int_{B(x, s)} |f^\kappa|^p\,dx &= s^{-\lambda} \int_{B(x,s)} \sum_{l=1}^\infty |\kappa(l)|^p s_l^{\lambda-n}\chi_l\,dx\\ &= s^{-\lambda} \sum_{l=1}^\infty \int_{B(x, s)} |\kappa(l)|^p s_l^{\lambda-n}\chi_l\,dx\\ &\leq s^{-\lambda} \sum_{l=1}^\infty s_l^{\lambda-n} \mu(B(x, s)\cap B(x_l, s_l))\\ &\leq s^{-\lambda}\sum_{l=1}^\infty s_l^{\lambda-n} \mu(B(x_l, s_l)) \\ &= s^{-\lambda}\sum_{l=1}^\infty s_l^{\lambda-n} \alpha_n s_l^{n} \\ &\leq \alpha_n s^{-\lambda}\sum_{l=1}^\infty \big(2^{-\lambda}\big)^l \\ &= \alpha_n s^{-\lambda} \frac{2^{-\lambda}}{1-2^{-\lambda}}\\ &\leq \alpha_n \big(\frac 1 2\big)^{-\lambda} \frac{2^{-\lambda}}{1-2^{-\lambda}} = \frac{\alpha_n}{1-2^{-\lambda}}, \end{align*} where the second equality uses the monotone convergence theorem, $\mu$ is the Lebesgue measure, and at the end we have used that $s_l=2^{-l}$, $\lambda>0$, and $s\geq \frac 1 2$. Note that the final bound is independent of $s\geq \frac 1 2$.
Now if $s<\frac 1 2$, then since $d(x_l, x_{l'})\geq 2$, $s_l=2^{-l}$, we see that $B(x, s)$ can only intersect at most one $B(x_l, s_l)$. Let this particular one $l$ be fixed in the next equation. Then when integrating over $B(x,s)$, only the term for this particular $l$ in (1) is nonzero. We have \begin{align} s^{-\lambda}\int_{B(x, s)} |f^\kappa|^p\,dx &= s^{-\lambda}\int_{B(x, s)} |\kappa(l)|^p s_l^{\lambda-n}\chi_l\,dx\\ &\leq s^{-\lambda}s_l^{\lambda-n}\mu(B(x, s)\cap B(x_l, s_l))\\ &\leq \begin{cases} s^{-\lambda}s_l^{\lambda-n}\mu(B(x, s))=s^{-\lambda}s_l^{\lambda-n}\alpha_n s^n = \alpha_n\big(\frac{s}{s_l}\big)^{n-\lambda}\leq \alpha_n & \text{if }s\leq s_l\\ s^{-\lambda}s_l^{\lambda-n}\mu(B(x_l, s_l))=s^{-\lambda}s_l^{\lambda-n}\alpha_n s_l^n = \alpha_n\big(\frac{s_l}{s}\big)^{\lambda}< \alpha_n & \text{if }s_l<s, \end{cases} \end{align} where we have used $0<\lambda<n$. Note that this bound for $s<\frac{1}{2}$ is also independent of $s.$ Therefore, $$ \|f^\kappa\|^p_{L^{p,\lambda}({\mathbb R}^n)} = \sup_{x\in {\mathbb R}^n, s>0} s^{-\lambda} \int_{B(x, s)} |f^\kappa|^p\,dx \leq \alpha_n\max\Big(\frac{1}{1-2^{-\lambda}}, 1\Big) = \frac{\alpha_n}{1-2^{-\lambda}}, $$ so $f^\kappa\in L^{p,\lambda}({\mathbb R}^n)$.
For a subset $I\subset {\mathbb N}$, define $\kappa^I:{\mathbb N}\to \{0,1\}$ by $$ \kappa^I(l) = \begin{cases} 1 & l\in I,\\ 0 & l\notin I.\end{cases} $$ Now let $I^1$ and $I^2$ be two distinct subsets of ${\mathbb N}$. Then there exists $l\in {\mathbb N}$ such that $|\kappa^{I^1}(l)-\kappa^{I^2}(l)|=1$. Then we have, by considering the ball $B(x_l, s_l)$ for this particular $l$, \begin{align*} \|f^{\kappa^{I^1}}-f^{\kappa^{I^2}}\|_{L^{p,\lambda}({\mathbb R}^n)} &\geq s_l^{-\frac{\lambda}{p}} \Big(\int_{B(x_l, s_l)} s_l^{\lambda-n}\,dx\Big)^\frac{1}{p}\\ &= s_l^{-\frac{\lambda}{p}} (s_l^{\lambda-n}\alpha_n s_l^n)^\frac{1}{p}\\ &= \alpha_n^{\frac{1}{p}}. \end{align*} Since the set of subsets $I\subset {\mathbb N}$ is uncountable, we have shown that this version of the Morrey space $L^{p, \lambda}({\mathbb R}^n)$ is not separable.