Let each line of a projective plane $\ P\ $ have $\ p\ge 5\ $ points. Let lines $\ L_0\,\ L_1\,\ L_2\ $ of $\ P\ $ have empty intersection $\ L_0\cap L_1\cap L_2\ =\ \emptyset.\ $ Define
$$ H\ :=\ P\setminus(L_0\cup L_1\cup L_2) $$
The H-lines are defined as sets $\ H\cap L,\ $ where $\ L\ $ is a projective line in $\ P.$
Then $\ H,\ $ together with H-lines, is a hyperbolic plane such that each H-line has $\ p-2\ $ or $\ p-3\ $ points, where both these cardinalities do happen.
More generally, one may consider any family of three or more projective lines (instead of three of them), which have an empty intersection. Then, with a bit of care, one gets (infinitely) more required examples with more than two different cardinalities of the H-lines.