The answer is: they are not the same. Here is a related question from another site where somebody asked what the difference between both of them is (found the sentence I asked about in the comments there too). This is the main reference my answer is based on.
"A vanishing point is a point on the image plane of a perspective drawing where the two-dimensional perspective projections (or drawings) of mutually parallel lines in three-dimensional space appear to converge."
The centre of projection (COP) is not on the image plane as you can see on the image you posted. It is the focal point where all your projection lines converge. If you place it on the image plane you would create a singularity and the whole projection math would fail.
However, the location of the vanishing point on the 2d plane depends on the position of your COP. Take the drawn cube on the projection plane of your image as an example. The top plane has 2 vanishing points because it has two sets of parallel lines. Now if you rotate the COP left or right around the 3d cube until one set of lines of the top face is parallel to the projection plane you would observe the following: One vanishing point would move directly to the 12 o'clock position while the other would accelerate away from the plane until it disappears because as you wrote:
parallel lines that are not parallel to the viewing plane converge to a vanishing point
Lines parallel to the projection plane do not have a vanishing point.
So how does it come that the internet says otherwise? My educated guess here is, that like with many other things, people start mixing up terms of closely related things. Also, you provided this link here as a reference. Strictly, they are not saying both are the same. They simply just name them identically. But that doesn't make it any better.
A noteworthy difference between COP and vanishing point (as mentioned by Wyck in the questions comments) is that the vanishing point is actually a 2d point while the COP is a 3d point. One can deduce it from my answer, but I felt like it should be mentioned explicitly.