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 the 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 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.