It is fundamentally a question of spectral lines.
Type I supernovae have no hydrogen lines, and type II have strong hydrogen lines. Type 1b have strong helium lines and no hydrogen lines, and type 1c have neither helium nor hydrogen lines. That’s why they aren’t designated as type II supernovae. The reason why type 1b and 1c supernovae have no hydrogen lines is because they’ve been stripped of the hydrogen of their outer envelope before core collapse (and in the case of 1c, their helium envelope as well).
They have this lack of hydrogen lines in common with type 1a supernovae, but distinctly lack silicon lines.
Naturally type 1b and 1c super novae will be smaller than a type II if they are imagined to have the same starting mass somehow pre-envelope loss, because there will be less material to collapse, and so less material to be thrown out.