The Wiener measure $W$ on the space of (continuous, a posteriori) curves defined on $[0,t]$ is uniquely characterized by being Borel and having prescribed pushforwards (that I shall not write here). It is immediate that $W$ is concentrated on the space of curves that start in $0 \in \mathbb R^n$ at time $0$.
What is the rigorous definition of the conditional Wiener measure?
Intuitively, I understand the conditional Wiener measure $W_p$ to be a Borel measure, with very similar pushforwards (I know them, I shall not write them here), but concentrated on the curves that also have the endpoint fixed: at time $t$ they arrive in $p \in \mathbb R^n$. It follows that $\int _{\mathbb R^n} W_p (A) \ \mathrm d p = W(A)$ for all Borel subsets $A$ of the space of curves. The problem is that this cannot be a definition, because nothing guarantees the uniqueness (and pointwise existence) of the disintegration $p \mapsto W_p$ of $W$ like in the formula above (the disintegration theorem does provide uniqueness almost everywhere, but under assumptions about the pushforward of $W$ that are definitely not met here).
(Please provide measure-theoretic explanations, not probabilistic ones, because I am not familiar with the probabilistic language.)
[In your integration formula, $dp$ should be replaced by $(2\pi t)^{-n/2}\exp(-|p|^2/2t)\,dp$.]
One way to present $W_p$ is as the image of $W$ under the transformation sending the path $\{x(s), 0\le s\le t\}$ to the path $[0,t] \ni s\mapsto x(s)-(s/t)[x(t)-p]$. This choice makes the disintegration formula true, and $p\mapsto W_p$ is weak${}^*$ continuous, not just measurable. As such it is unique.
By the way, as the space of continuous functions mapping $[0,t]$ to $\Bbb R^n$ is Polish, a standard disintegration theorem does apply to yield an a.e. determined family $\{W_p: p\in\Bbb R^n\}$. This general result only ensures that $p\mapsto W_p(A)$ is Borel for Borel $A$.