Re: names of angle systems
I'd call the first one E-centric, but only to make a pun. I've never actually heard of either one.
Re: names of angle systems
If you meant to have some negatives in the first system, it's the one used in the complex plane; in trig, angles of that form are called "reference angles". The second system is the second common parametrization of a circle, though I don't have a formal name for it either.
I suspect that names either don't exist or would be unhelpful to your audience, but I don't know your application.