Ok to answer your questions one by one:
Usually the term chin-up is used when your palms are facing
forward. Pull-up is used when your palms are facing you. Therefore
you got the terms correct.
Chin-ups and pull-ups involve quite a number of muscles to
perform. Some are used a lot, the others a little. The main ones
are your back muscles and biceps. To a lesser extent,
your abs, shoulders and maybe some minor muscle groups. There was
once I trained chin-ups so much that even my chest and abs hurt the
next day. So, you're not wrong to say that abs and chin/pull-ups
are linked.
Chin-ups use more of your lats while pull-ups use more of your
biceps. Of course, there are ways of varying the way you do them to
make more use of certain muscle groups.
I won't recommend doing push-ups because they work mainly the
triceps, chest and shoulders. These muscles are not used as much as
the main ones that I mentioned: biceps and back. I know people who
can do 50 standard push-ups in one go and yet cannot even do one
chin/pull-up. Unless you have plenty of time to train, aim for the
main muscles for chin/pull-up first.
The various ways of doing push-ups simply varies the effort
focus on chest or triceps. Try all of them to get an all-rounded
workout for your arms and chest.
There shouldn't be much of a difference between a tall or short
bar other than the convenience of relaxing your legs on a tall
bar.
How many do you need to pass your test? Assuming you need 6, and
you can only do 4.5, my suggestion is to do 3 sets of 3-4 with a
short rest in between. Don't do to failure. Doing to failure will
not do much for strength building. Rinse and repeat.
In the meantime while you are training your chin/pull-up, don't
forget the rest of your stations. Run frequently for your 2.4km and
do your crunches (and your push-ups if you have extra time).
All-rounded fitness should help your chin/pull-up training.
