Motivation
I am trying to figure out what the intrinsic definition of different variants of "tensors" are supposed to be. I managed to find the transformation rules for the coefficients of the above objects on wikipedia, but I cannot find an intrinsic definition for all of those, so I want someone to double check my understanding. I am trying to better understand those in order to comprehend what is going on with the whole densitization and pseudofication busines with differential forms, densities, and pseudodifferential forms. Ultimately I am trying to understand what we were actually integrating in basic calculus - i.e. was it differential forms, differential density forms, pseudotensors, densities as in measure theory, or something else. I thought I would study this first in a simpler setting, where I just have constant valued tensors without any mention of manifolds and tangent spaces. I added the above just as a motivation for my question, the question is still very much about how one defines different "variants of tensors" mathematically, rather than just stating the rules by which their coefficients transform under a change of basis.
Tensors
Wikipedia says that given a vector space $V$ a $(p,q)$ tensor $t$ is an element of the tensor product space: $$T^p_q(V) = \underbrace{V\otimes \ldots \otimes V}_p\otimes \underbrace{V^*\otimes\ldots\otimes V^*}_q = V^{\otimes_p}\otimes (V^*)^{\otimes_q}.$$
Given a basis $B=(e_1,\ldots,e_n)$ for $V$, I can construct the unique biorthogonal basis $B^*=(e^1,\ldots,e^n)$ of $V^*$ such that $e^i(e_j) = \delta^i_j$, and I will also use $e_j(e^i) = \delta^i_j$ by $V^{**}\cong V$. Then define $$([t]_B)^{i_1\ldots i_p}_{j_1\ldots j_q} = t(e^{i_1}\otimes\ldots\otimes e^{i_p}\otimes e_{j_1}\otimes\ldots\otimes e_{j_q}), \quad 1\leq i_k, j_l \leq n, \,\, 1\leq k\leq p,\,\, 1\leq l\leq q.$$ That is, I get $n^{(p+q)}$ coefficients arranged in an $(p+q)$-dimensional array. I can now represent $t$ as: $$t = \sum_{i_1=1,\ldots,i_p=1}^{n,\ldots,n} \sum_{j_1=1,\ldots,j_q=1}^{n,\ldots,n}([t]_B)^{i_1\ldots i_p}_{j_1\ldots j_q} e_{i_1}\otimes \ldots \otimes e_{i_p} \otimes e^{j_1}\otimes\ldots\otimes e^{j_q}.$$ Now let $C = (f_1,\ldots,f_n)$ be a basis such that $e_i = \sum_{i=1}^n f_kP^k_i$, I get: \begin{equation} \begin{aligned} t &= \sum_{K,L}\underbrace{\left(\sum_{I,J}([t]_B)^{i_1\ldots i_p}_{j_1\ldots j_q} P^{k_1}_{i_1}\ldots P^{k_p}_{i_p}(P^{-1})^{j_1}_{l_1}\ldots (P^{-1})^{j_q}_{l_q}\right)}_{([t]_C)^{k_1\ldots k_p}_{l_1\ldots l_q}} f_{k_1}\otimes \ldots \otimes f_{k_p} \otimes f^{l_1}\otimes\ldots\otimes f^{l_q} \\ &= \sum_{K,L} ([t]_C)^{k_1\ldots k_p}_{l_1\ldots l_q} f_{k_1}\otimes \ldots \otimes f_{k_p} \otimes f^{l_1}\otimes\ldots\otimes f^{l_q}. \end{aligned} \end{equation} Thus I could have just started from the space of multilinear functions: $$L(\underbrace{V^*,\ldots,V^*}_p,\underbrace{V,\ldots,V}_q;\mathbb{F}) \cong T^p_q(V),$$ and the transformation rules would have been a natural consequence. Of course, conversely, any $n^{p+q}$ array of numbers that transforms in the above way is the representation of a tensor in a specific basis.
Tensor Densities
I couldn't find an intrinsic definition of tensor densities, but there's an example of the transformation rules: $$([t]_C)_{k}^{l}=(\det P^{-1})^s\sum_{i,j}([t]_B)^{i}_{j}P^{k}_{i}(P^{-1})^{j}_{l}.$$ I also believe that alternating forms transform in the same way, so it would seem that if $s>0$ this is really a $(p,q+s\cdot n)$ tensor formed as: $$T^p_q(V)\otimes (\operatorname{Alt}^0_n(V))^{\otimes_s} \cong T^p_q(V) \otimes ({\textstyle\bigwedge}^nV^*)^{\otimes_s},$$ where $\operatorname{Alt}^a_b(V)$ is the space of antisymmetric tensors of valence $(a,b)$ over $V$ and ${\textstyle\bigwedge}^nV^*$ is the $n$-the exterior power of the dual. I believe that in order to get the determinant the specific form of the alternating tensor ought to be: $$d_n = \sum_{l_1,\ldots,l_n}\epsilon_{l_1\ldots l_n}e^{l_1}\otimes\ldots\otimes e^{l_n} = \sum_{\sigma\in S_n}\operatorname{sign}(\sigma)e^{\sigma(1)}\otimes\ldots \otimes e^{\sigma(n)},$$ where $\epsilon$ is the Levi-Civita symbol. Similarly for $s<0$, I expect to get a $(p-s\cdot n,q)$ from: $$T^p_q(V)\otimes (\operatorname{Alt}^n_0(V))^{\otimes_{-s}}\cong T^p_q(V)\otimes (\Lambda^n(V))^{\otimes_{-s}},$$ with the representative from $\operatorname{Alt}^n_0(V)$ being: $$d^n = \sum_{k_1,\ldots,k_n}\epsilon_{k_1\ldots k_n} e_{k_1}\otimes\ldots\otimes e_{k_n}.$$ Finally, the multilinear objects that transform in this way are not only the above. Let $s = b-a$, then I can form $(p+a\cdot n, q+b\cdot n)$ tensors: $$T^p_q(V)\otimes (\operatorname{Alt}^n_0(V))^{\otimes_{a}}\otimes(\operatorname{Alt}^0_n(V))^{\otimes_{b}}\cong T^p_q(V)\otimes (\Lambda^n(V))^{\otimes_{a}}\otimes (\Lambda^n(V^*))^{\otimes_{b}},$$ which by the transformation rules definition is an $s$-density since $$(\det P)^a(\det P^{-1})^b = (\det P^{-1})^{b-a} = (\det P^{-1})^s.$$ The main difference to just specifying the transformation rules is that now I know that those transformation rules define a whole equivalence class of tensors such that two tensors are within the class of the $s$-densities if $b-a=s, \, a,b\geq 0$. Furthermore, the multilinear map counterpart seems to be: $$T^p_q(V)\otimes (\operatorname{Alt}^n_0(V))^{\otimes_{a}}\otimes(\operatorname{Alt}^0_n(V))^{\otimes_{b}} \cong L(\underbrace{V^*,\ldots,V^*}_p,\underbrace{V,\ldots,V}_q;(\Lambda^n V)^{\otimes_a}\otimes (\Lambda^n V^*)^{\otimes_b}).$$ This means that instead of $\mathbb{F}$ the map returns a tensor $(\Lambda^n V)^{\otimes_a}\otimes (\Lambda^n V^*)^{\otimes_b}$.
Pseudotensors
A pseudotensor is defined here to transform like a usual tensor but also gains a sign from the determinant of the transformation. The only ways I can think of achieving this is by considering spaces of the forms:
$$T^p_q(V)\otimes |(\operatorname{Alt}^{n}_0(V))^{\otimes_s}|\otimes (\operatorname{Alt}^{0}_n(V))^{\otimes_s}, \quad T^p_q(V)\otimes (\operatorname{Alt}^{n}_0(V))^{\otimes_s}\otimes |(\operatorname{Alt}^{0}_n(V))^{\otimes_s}|,$$ where $s>0$ is odd. The absolute value is to be understood as being applied at the end: $$d_n\in |(\operatorname{Alt}^{0}_n(V))| \implies d_n = \left|\sum_{l_1,\ldots,l_n}\epsilon_{l_1\ldots l_n}e^{l_1}\otimes\ldots\otimes e^{l_n}\right|.$$
This is because in the two cases we will get extra factors that cancel up to sign: $$|(\det P)^s|(\det P^{-1})^s = \operatorname{sign}(\det P), \quad (\det P)^s|(\det P^{-1})^s| = \operatorname{sign}(\det P).$$
Pseudotensor Density
A pseudotensor density with weight $s$ supposedly transforms with with the $s$-th power of the determinant multiplied by the sign. I do this by mixing the ideas from the density and pseudotensor formulation, using $s = b-a, \, a,b\geq 0$: $$T^p_q(V)\otimes |(\operatorname{Alt}^{n}_0(V))^{\otimes_{1+a}}|\otimes (\operatorname{Alt}^{0}_n(V))^{\otimes_{1+b}}, \quad T^p_q(V)\otimes (\operatorname{Alt}^{n}_0(V))^{\otimes_{1+a}}\otimes |(\operatorname{Alt}^{0}_n(V))^{\otimes_{1+b}}|,$$
Even and Odd Tensor Densities
For integer even weights $s$, people seem to term densities "even densities", and pseudodensities "odd densities". Vice versa for $s$ odd, densities are termed "odd densities", and pseudo densities "even densities". Basically pseudo densities are termed with the opposite parity of $s$, while densities are termed with the parity of $s$. For even tensors with non-integer weights $s$ one could take powers of the absolute value, e.g. $|(\operatorname{Alt}^{0}_n(V))^{\otimes_{1+b}}|^s$, and once again this is not the only way to get such tensors.
Pseudoscalar and Orientation
I have seen people explaining that one could build pseudotensors (i.e. the ones I required a nasty absolute value for, which broke multilinearity) from tensors by multiplying them with a pseudoscalar. I saw e.g. that it can be defined in a Clifford algebra, but I was hoping there was a formulation that just sticks with tensor products and uses the minimal possible mathematical formulation. As far as I understood the pseudoscalar behaves similar to orientation when integrating on manifolds. I found some notes that try to define this object, but it's pretty hard for me to figure out what the author meant exactly. He introduces the two possible orientations formally as a set $o_V = \{a,b\}$ and extends it to an algebra $\{-1,1,a,b\}$ such that $a^2=b^2=1$ and $a=-b$. Then on page 29 he defines the space $\psi V = \operatorname{span}(a) = \operatorname{span}(b) = \{\mathbb{R}\cdot a\}$ of pure pseudo-scalars. And the space of mixed pseudo-scalars $\psi^* V = \mathbb{R}\oplus \psi V$. From those he generates pseudo-tensors by taking the tensor product $T^k_l(V)\otimes \psi V$, and in order to support both normal tensors and pseudotensors he also considers $T^k_l(V)\otimes \psi^* V$, which through multilinearity is isomorphic to $T^k_l(V\otimes \psi^* V)$.
I was wondering for a while why $\psi V$ depends on $V$, or rather why $o_V = \{a,b\}$ depends on $V$. However later on a function $o$ appears - I either missed its definition, or it just popped out. My understanding is that it accepts the coordinate functions in his case and returns an orientation based on those. I cannot reconcile that orientation function with the one here however: $o(Av_1,\ldots,Av_n) = \operatorname{sign}(A)o(v_1,\ldots,v_n)$, which seems to be much more coherent as it takes the basis vectors as arguments. In either case, in the latter this set forms a linear space where the two elements such that $|o(v_1,\ldots,v_n)|=1$ are termed the orientations. Are the $a$ and $b$ from $o_V$ the two classes from the set of functions $o$? Then would I be able to define a pseudo-tensor as: $$t = o(e_1,\ldots,e_n)\sum_{I,J}T^{i_1\ldots i_p}_{j_1\ldots j_q} e_{i_1}\otimes\ldots\otimes e_{i_p}\otimes e^{j_1}\otimes \ldots \otimes e^{j_q}?$$ How do I even work with this $o(e_1,\ldots,e_n)$ in practice? When I change coordinates do I just write $o(e_1,\ldots,e_n) = o(Pf_1,\ldots,Pf_n) = \operatorname{sign}(\det P)o(f_1,\ldots,f_n)$ and then just go about my day? And I guess if I ever multiply this with another pseudo-tensor involving $o$ it will cancel out?