This is very good, I did intend to use it in the setting of algebraic varieties, so the use of degree one polynomial feels quite natural.

I was talking about homotopy continuation on an algebraic variety. So eventually I must use the last two definitions. But those, I thought, are too abstract, so I wanted to start with something more intuitive, i.e., the first definition. I didn't have to show that they are all equivalent, but that'd be an obvious question to ask.

I'm not sure if I understand this. I know the algebraic closure of the ring of formal Laurent series is the formal Puiseux series, and this form a basis for the formal local parametrization of plane algebraic curves. But I still don't see how this extends to the case of space algebraic curves, unless I'm missing something. I just did a quick check on Serre's Local Fields (my French is, uh..., terrible), and I couldn't find it. Do you happen to remember which chapter is it in? Either way, a purely algebraic proof is probably not enough, as it require the concept of convergences.