In this lecture by Prof. F. P. Schuller (I've included the correct time stamp) it is claimed that two tangent spaces at different points are disjoint, i.e. $T_pM\cap T_qM =\emptyset$ for $p,q\in M$ and $p\neq q$. In this lecture, the tangent space at a point $p\in M$ was defined as
$$ T_pM:=\{X_{\gamma,p}|\gamma \,\mathrm{is\, a\, smooth\, curve\, through\,p}\} \quad,$$
i.e. the collection of the directional derivatives along all (smooth) curves through $p$:
$$X_{\gamma,p}:\mathcal C^\infty(M) \longrightarrow \mathbb R$$ with
$$ X_{\gamma,p} : f\mapsto \left(f\circ\gamma\right)^\prime(0) \quad ,$$
where $\gamma(0)=p$.
I don't understand the aforementioned claim. If we take two constants curves $\gamma_1$ and $\gamma_2$ going through $p$ and $q$, respectively then we have two directional derivative operators at these points which are both the $0$-map, i.e. $X_{\gamma_1,p} = X_{\gamma_2,p}=0$, where $0: f\mapsto 0 $ by a mild abuse of notation; so I think that since both are equal as operators, both are elements in each tangent space. What do I miss here? Further, is there any non-trivial "overlap" between two tangent spaces at different points?
PS: I know there are some other question regarding the disjointness of tangent spaces at different points here on math stackexchange; however, I could not find the definition used here.
Thanks to the very helpful comments of @Didier, I think I understand the issue. The following answer just summarizes the comments. If someone still wants to expand and give a more elaborated explanation, please do so - I'd highly appreciate it.
Indeed, with the definition of $T_pM$ as in the lecture the tangent spaces are not disjoint, cf. my counter example in the question. For our purposes, the construction of the tangent bundle $TM:=\dot \bigcup_{p\in M}\,T_pM$, however, this causes no problems since we consider the disjoint union of the tangent spaces.
Yet, we can construct tangent spaces which are mutually disjoint. To do so, we have to build the tangent spaces at a point $p\in M$ from operators defined on the germs of (smooth) functions at $p$. By doing so, the tangent spaces at different points are mutually disjoint because their elements are defined on different domains.