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.
Let me present a proof of property 4: INDEED, let $\ x\in H\setminus L,\ $ where L is an H-line, and let $\ L'\ $ be the projective extension of $\ L.\ $ For each $\ y\in L'\cap M',\ $ where $\ M'\ $ is one of the removed projective lines, we get the H-line $\ xy\cap H\ $ which is parallel to $\ L,\ $ where $\ xy\ $ is the projective line passing through $\ x\ $ and $\ y.$
There is also a removed line $\ N'\ $ which doesn't pass through $\ y\ $ (the intersection of all removed lines is assumed to be empty). Thus $\ \exists_z\ y\ne z\in L'\cap N',\ $ and $\ xz\cap H\ $ is another H-line parallel to $\ L\ $ different from the previous parallel line.
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.
