For a graph to be an euler graph, all the vertices have to be of even degree...I was wondering "does which vertex to start matter?"...I played around and now I kind of think it doesn't matter where you start...you can always end up at the initial vertex itself...because degree of every vertex is even...so if we go to a vertex we need to exit it either and in the end when we are out of non-visited edges, the only option will be to go to the vertex we started.
Am I thinking it the right way?
Yes you are exactly right. When I was young, instead of graphs, I thought about what drawings you can make without lifting your pen from the paper. You see the obvious relation.
There are two types of drawing that can be done (so most can't): the ones where every point of the drawing touching or crossing itself has an even number of linesegments entering or leaving and the drawings where the number of points with an odd number of segments is exactly 2. The famous house with a cross is the standard example of that type. In that type it does matter where you start (and the reasoning is the same): you start at one of the two odd vertices and end at the other