I want a solenoid that stays retracted and propels forward when current is passed through it. Is there a solenoid that does this? What is the difference between push and pull solenoid?
Solenoids are just coils of electrical wire with a moving armature that wants to get into the point of lowers magnetic reluctance.
All the push solenoids I have seen basically have a mechanical arrangement that delivers the force out through the bottom of what would typically be the closed end of a pull solenoid housing. It could be a thinner push-rod or a full size rod or pipe made of a non- or less magnetic material so it does not get attracted into the solenoid like the main armature.
It is also possible to get bistable solenoids that have permanent magnetic parts, these may have a pair of windings or be polarised. I suppose one could get a solenoid with a permanent magnet armature that you force to push out with a determined polarity DC signal.
EDIT: The Ledex list of tubular solenoids includes about one third in the push configuration, even so the pull ones probably outsell the push ones at over 10 to one at a guess. You should be able to find at least some of the popular types at most industry outlets.
There are some good descriptions in this other Q&A