What is the fiber of the path space fibration? Schematically, I understand the path space fibration $PX$ over some path-connected, pointed topological space $X$ with base point $x_o$ as:
$$\Omega X \hookrightarrow PX \twoheadrightarrow X,$$
where the first arrow is the inclusion and the second is the evaluation map. 
If we understand $PX$ as the space of paths in $X$ (i.e., continuous maps $p(t)$ from the unit interval to $X$, with $p(0) = x_o$), then it seems that the natural definition of $\Omega X$ is the space of paths $p(t)$ with $p(0) = p(1) = x_o$. However, I have found conflicting statements from various sources:


*

*This MSE question says that $\Omega X$ is the space I have just described.

*The Wikipedia article for path space fibration says that $\Omega X$ is the loop space of $X$; the Wikipedia article for loop space defines that space as the space of continuous pointed maps from $S^1$ (with base point) to $X$.

*This MSE question points out that the space of continuous pointed maps from $S^1$ to $X$ is not the same as the space of paths in $X$ beginning and ending at the same point, which I think is correct due to the fact that elements of the latter may be discontinuous at $x_o$ between $t=1$ and $t=0$.


Having become thoroughly confused by everything I found on the internet, I turned to Hatcher, where I discovered a lengthy and fairly dense discussion on loop spaces which requires the use of "James reduced products," a concept I have not encountered before in my coursework. This leads me to believe that perhaps the whole situation is more complicated than I initially thought.
My question is, what is $\Omega X$, and what subtlety am I missing here that is leading to the confusion described above?
 A: The definition in your question is that for pointed spaces $(X,x_0)$. More precisely we should write $P(X,x_0) = (X,x_0)^{(I,0)}$ = set of all basepoint-preserving maps $(I,0) \to (X,x_0)$ with compact-open topology for the pointed path space, $p : P(X,x_0) \to (X,x_0), p(u) = u(1)$, and $\Omega(X,x_0) = p^{-1}(x_0)$ fot the pointed loop space. Both have as basepoint the constant path at $x_0$. Then $\Omega(X,x_0)$ is the fiber over the basepoint $x_0 \in X$.
You can do an analogous construction for unbased spaces $X$:
$PX = X^I$ = set of all maps $I \to X$ with compact-open topology is the free path space. The evaluation map $p : PX \to X, p(u) = u(1)$, is a fibration. Its fibers are the sets $p^{-1}(x) = \{u \in X^I \mid p(u) = u(1) = x \} =(X,x)^{(I,1)}$. The latter is homeomorphic to $P(X,x)$.
A third construction is the free loop space of a space $X$:
$$\mathcal L X = X^{S^1} .$$
It can be viewed as the unpointed version of $\Omega (X,x_0)$. There is a  canonical embedding $\iota : \Omega (X,x_0) \to \mathcal L X$: Each $u \in \Omega (X,x_0)$ is a closed path $u : I \to X$ such that $p(0) = p(1) = x_0$ which determines a unique continuous $\hat u : I/\{0, 1\} \to X$ and via the identification $I/\{0, 1\} = S^1$ this gives us $\iota(u) \in \mathcal L X$.
Note that this construction also allows to identify $\Omega (X,x_0)$  with $(X,x_0)^{(S^1,*)}$. In fact, $\iota(\Omega (X,x_0)) = (X,x_0)^{(S^1,*)} \subset X^{S^1}$.
