(BTW the Mathworld article on hyperbolic geometry makes the
oddly imprecise definition that in a hyperbolic geometry *many*
distinct lines pass thru P and are parallel to l. This offends my
nascent mathematical susceptibilities).

So which is hyperbolic - infinitely many, or > 2, or 'many', and
pls supply a def. of 'many' - yeah I know, 'many' means '> 2'.
Isn't 'many' generally to be avoided in mathematics?

Secondly, the three types of geometry define the case for all lines l,
and
for each line l, all points P not on it, belonging to a geometry of
that
type.

There is now I discover 'absolute' geometry which does not
invoke ANY version of the parallel postulate.

Am I correct in assuming that an absolute geometry allows for
parallel lines, but makes no universal claims relative to
lines l, points P not on them, and lines passing thru P
and possibly parallel to l? So there could be
lines l and points P with 0, 1, 2, or oo many lines passing
thru P parallel to l, all in the same geometry?