When a touching ball has been called, the striker must play the cueball away from that ball without moving it or it is a push stroke.
Snooker is a whole different animal and probably shouldn't even be brought up.
In snooker you flat out can't play the frozen ball, you cannot budge it even a millimeter.
If, for example, you shot almost parallel to the tangent line, thinly grazing the object ball as the cue ball
rolls away, they label it a "push stroke" even though it absolutely is not a push, it's just a normal stroke
and no double hit has occurred.
In pool this shot is entirely legal, frozen or not.

For practical purposes, a push shot is simply a double hit that happens so fast that it's hard to see.
Though there are weird shots where the cue tip can stay on the cue ball a long time (like when an OB
is wedged between the CB and rail) and that's also a considered a push. But that's very rare.
99% of the time when people cry "push!", they're saying the shooter double hit the cue ball.
In theory, if there's no gap, it's not considered a push in pool, and you can shoot straight into the OB.
Both balls scurry away from the tip simultaneously.
However a double hit is still possible, if the tip is still accelerating and the CB doesn't move away
quickly enough, your follow through can hit the CB again and give it a little extra boost.
You don't see the second hit but the rapid rolling action of the cue ball gives it away.
I suspect the snooker rule was implemented to avoid arguments about whether a double hit may have
occurred anyway. Whereas in pool we let the ref suffer through making difficult judgment calls like
this one: http://www.youtube.com/watch?feature=player_detailpage&v=bz91kJcVB7A#t=113s
In pool, each tournament has 2 way of handling frozen cue balls.
Version A: the shooter must not shoot directly into the frozen balls, he must shoot away at an angle
(similar to my example image above)
Version B: the shooter can shoot any way he wants into the frozen balls, even straight at them.
To reiterate what someone else pointed out earlier. Jeremy's complaint was NOT that he felt
Shane pushed (aka double hit) the cue ball. His complaint is that he thought the tournament
was using "version A", but after shane's push, the ref claimed they were going by "version B".
Jeremy apparently felt like they changed the rule mid-tournament or ignored it just to humor shane.
As for the whole ball cleaning thing... if you watch TAR's stream and pay attention, those balls
are highly polished and the big black scuff marks caused by diamond's pockets are obvious,
you can see these rubbery black skid marks on several object balls.
Good players know that hitting that mark can definitely alter the path of the object ball,
and cause a perfectly aimed shot to miss.
I'm sure some of you have played 50 years without ever cleaning the balls midrack.
And I'm also sure you had many balls skid on you during that time
and you either didn't know or didn't care. You're playing for fun, or you're playing a set,
or you're playing joebob's weekly bar tournament. If a skid costs you the match, there's
more matches next week and you can just hit up the ATM. So it's no biggie.
This is a major tournament that comes around just once a year. If a skid costs you the match,
you're out and you won't get another shot for 12 months. So yeah, players are gonna take precautions
and ask for balls to be cleaned. It's not nitpicking, it's necessary.