Compute fundamental group _visually_ by the polygonal representation of the space

Maybe this is the picture you are looking for. I refer to a path tracing the oriented edge as $a$, and its inverse as $a^{-1}$. Note that every continuous path on the dunce cap (any CW-complex really) can be continuously deformed to trace only edges (of the given CW-structure). This should be somewhat visually intuitive, by straightening your path along edges, vertex to vertex. Thus if the path $a$ is trivial, so is any other path (do you see why?).

A comment as to what deformations are allowed - you are free to push your path anyway you want, and over edges (reemerging on any of the copies of the edge, noting orientations). This is what happens between the first and the second image. The one important rule is that you CANNOT ever move the basepoint (=initial and final point of your path).

Note that I did not move the basepoint passing from the first to the second picture, as the three corners of the triangle are literally the same point of the dunce cap.

Edit: As per your edit, one way to do it directly is this. Usually, you would just say the dunce cap is connected, thus its fundamental group relative to some basepoint is trivial if and only if it is trivial for some other basepoint (which we have shown above).