By the way, the definition with tubular neighborhoods, allows for two isometric triangles to have interiors with arbitrary topology, and so two isometric triangles homologous to zero can bound interiors with completely different area.

Ah, no problem. It's been an interesting discussion. Plus I'm not even sure which term is better in this situation.

I think when people say congruent they mean global isometry, like a ''motion" of the whole space (e.g. hyperbolic plane) that moves one triangle onto the other, so that they match precisely. It is a notion related to the isometry group of the whole space (hence Igor's answer). The term isometry, however, I think means something more general and can be used in a more flexible way: a map that preserves lengths (and angles when angles are well defined). Hence my definition of isometric triangular piece-wise geodesic curves (we call triangles) in terms of annular/tubular/collar/ neighborhoods.

Yeah, it seems to me that your comment confirms my assumption that what I wrote after the edit is what you are probably interested in. I basically tried to put it in a more ''formal" way. I am sorry if I have been a bit sloppy and not too precise.

Yes, I think $(x^2+y^2−1)(x^2+y^2−4)−1$ has a singularity at infinity, so as a compactified Riemann surface it is singular. First take $y \mapsto iy$ to rotate the points at infinity in the real projective plane and then perform the projective transformation $x \mapsto 1/x$ and $y \mapsto y/x$. Then you see two ovals touching at two points. In my example this does not occur because it is ultra-Morse: its leading highest order term is of the form $(x-a_1y)...(x-a_ny) + \text{lower order terms}$ which makes it non-singular at infinity for different $a_j$.