There's 3 flavors of "push" which is why it's the source of much arguing and confusion.
Two of them are generally considered fouls, and one is not.
1. CB not frozen to the OB.
The cue tip hits the CB, which travels forward a bit and then hits the object ball.
A few milliseconds later, in the same stroke, the tip hits the CB a 2nd time.
This is a double hit, and a double hit is always a foul.
Usually, this foul is obvious if you look at the speed of the cue ball after the stroke.
It will move forward without pausing, at nearly the same speed as the object ball did.
If it were moving forward from ONLY topspin, and not from a double hit,
then its speed would be much lower than the object ball's speed.
Occasionally it's not obvious, if the player elevates.
Elevating is a trick you can use to avoid double hitting, but it doesn't always work.
When it fails, you will see the CB sort of dive forward and then die,
because the stick clipped the back of the CB before it could scoot away.
You can see a video of this foul here. It's tough to call this one on someone, especially if the cue ball
only goes forward a few inches.
http://www.youtube.com/watch?v=aQpYo2uv-rU&t=75
To further complicate things, if you elevate and
actually jump the cue ball,
it's possible for the cue ball to travel forward a little bit,
even if there's no second hit.
Because the CB is hitting the top of the object ball and continuing forward while airborne.
Then when it lands, any spin you applied will take effect.
Including topspin which might make it go further forward.
So if you elevate to avoid a close hit, and jump the cue ball, you might make
a perfectly good hit, but it might
look like a bad double hit.
2. CB frozen to the OB.
Technically you can push straight through both balls in this situation, as there's no double hit.
The cue ball action looks almost exactly like it does in an illegal push shot,
so the only way to be sure is confirm BEFORE the shot of they're frozen or not.
3. 'Extended contact' push.
This is not a double hit, but it's still a foul, you are not allowed to keep the cue tip
in extended contact with the object ball in order to, for example, do a trick shot like the one in the OP's video.
In pool you must
strike balls with the cue stick, not shove or steer them with it.
WPA rules:
1.
If the cue stick contacts the cue ball more than once on a shot, the shot is a foul.
2.
if the cue ball is touching an object ball at the start of the shot, it is legal to shoot towards or partly into that ball
3.
It is a foul to prolong tip-to-cue-ball contact beyond that seen in normal shots.
(this is the one that applies to the OP's question)
BCAPL (league) rules:
1.
It is a foul if your cue tip strikes the cue ball more than once on a single shot.
2.
If the cue ball is frozen to a legal object ball, it is legal to shoot toward the object ball
provided you use a legal stroke.
3.
It is a foul if you shoot a push shot.
A push shot is defined earlier in the rulebook as "
A shot in which the cue tip maintains contact
with the cue ball longer than the momentary contact allowed for a legal stroke."
BCA rules (non-league) - Exactly the same as the WPA rules.
APA rules
1.
A double hit is an illegal shot involving the tip of the cue stick
coming into contact with the cue ball twice during the execution of a single shot.
2.
A push shot involves a situation where the cue ball is frozen to the object ball.
The problem faced by the shooter is to keep from pushing or keeping the tip of the cue on the cue ball.
It looks bad and is generally thought of as illegal. Push shots are controversial.
Push shots will not be called in this amateur League.
Players who repeatedly guide the cue ball with force through object balls that are frozen to the cue ball,
using a level cue and long follow through, may be subject to a sportsmanship penalty.
To me this sends mixed messages. First they bunch together legal pushes
(the balls are frozen) with illegal 'extended contact' pushes.
But, arguably, a frozen ball will situation where you shoot into the object ball,
may result in extended contact even if you didn't do it on purpose.
So maybe they SHOULD be mentioned in the same paragraph.
The APA will not call EITHER type of push a foul, but they say
"if you do it a lot you might get called for as sportsmanship penalty".
I feel this is dumb, something either is a foul, or it's not.
There should be no such thing as "this isn't a foul unless you do it a lot."
You judge each shot on its own merits.
3.
See above.
tl;dr:
WPA = foul
BCAPL = foul
BCA = foul
APA = no foul unless you do it all the damn time.