"Induced EMF is magnetic, or from an electric field" is the distinction important in analysis? Their nature of induction is different, however, from relativity it seems that the two can be the same magnitude, in analysis of models relevant to EMF-induction is it important to identify that difference? That the $\epsilon $ is magnetic or from the electric field?
 A: It depends on what units you are using. Sometimes you can make $\mu$ and $\epsilon$ to have the same dimensions (or to be dimensionless Gaussian units), but this is a long long debate, regarding units. In my opinion, SI is better because even if you have extra factors and strange dimensionful constants you at least don't risk the run to not know what you're talking about. I urge you to read  https://doi.org/10.1119/1.1992863. I am positive that this very paper will make you understand the difference between Electric & Magnetic units. Compare Equation (3) and equation (8). They are quite a different beast. If after reading the article this thing troubles you, I suggest looking on Jackson's appendix, he dedicates an entire section to this and has a lot of articles as references but I warn you, this thing is and will never be settled.
A: I think you're asking about the difference between "motional EMF" (which originates from the Lorentz magnetic force) and "transformer EMF" (which originates from an electric field) (see here for example).
Faraday's law (in the form EMF = -d(flux)/dt) include both of these, and in practice most of the time that someone calculates EMF using Faraday's law, they don't know or care or think about whether it's motional or transformer EMF or both that they're calculating.
So, if you're an engineer deciding whether your electric generator should have a spinning magnet and stationary wire (transformer EMF) or a stationary magnet and spinning wire (motional EMF), you would consider the effects on efficiency and friction and lots of other things, but you wouldn't decide on the basis of what type of EMF it is creating. That's just not relevant information for you.
As you mention in the question, yes it's reference-frame dependent. In one inertial frame of reference, an EMF can be motional, but in another, the same EMF may be transformer. See here for example.
