What is the difference between semi-relativistic and non-relativistic limit of an equation?

I don't think it makes sense to talk about a "semi-relativistic limit."

In the non-relativistic limit, as you say, we have $v/c ll 1$, and we can use binomial expansion on the Lorentz factor to say things like

$$ E = gamma mc^2 = mc^2 times left( 1 + frac12frac{v^2}{c^2} + cdots right) approx mc^2 + frac12 mv^2 tag{slow} $$

In the ultra-relativistic limit, we have $gamma gg 1$, and we can say things like

$$ E^2 = (pc)^2 + (mc^2)^2 approx (pc)^2 tag{fast} $$

But if you're not working in either of these limits (if, for instance, you're in the regime where $gammasimeq2$ or $gammasimeq 10$), the neither of these approximations is correct or useful, and you have to keep track of all the $gamma$s rather than using one of the above tricks to make them go away. I have probably referred to this region as "semi-relativistic" in order to help my collider-physics colleagues remember that electrons do have mass. But it doesn't make sense to refer to this intermediate range as a "limit." It's just relativity.

If you're a three-significant-figures person, you might think "semi-relativistic" rather than the fast or slow limit for the interval $1.01 < gamma < 100$. Sometimes it's nice to think about $betagamma = gamma v/c$ over this intermediate range, since the behavior of that product doesn't run up against an asymptote like $beta$ or $gamma$ do.