In most American rocket launches, the "counter" will say:
3,2,1 [awkward pause], and liftoff....
So what truly happens at T-0? Is there any standardized event that this time indicates?
Is the pause only the delay between engine's ignition and moment of liftoff?
T minus zero appears to generally indicate the moment of booster liftoff, but can also indicate something different. To borrow Cort Ammon's wording, it's an action or event that results in a substantial change in the amount of control you have over the situation.
Taking the countdown for the Falcon 9 as an example, T-00:00:00 is the instant of lift-off; in other words, when the rocket has started rising ever so slightly off the launch pad.
There are other alternatives. For example, the Apollo Saturn V launch sequence had first stage engine ignition beginning at T minus 8.9 seconds, and the actual engine ignition procedure took almost a second. There, T minus 0 indicated release of the booster hold-down arms. If I recall correctly, the hold-down arms release was required to be simultaneous to the tune of something like 50 ms, to ensure that the rocket stayed upright. Releasing the hold-down arms allowed the rocket to start climbing, but wasn't by itself the definition of liftoff.
The Space Shuttle launch sequence had the main engine ignition beginning at T-6.6 seconds, launch thrust required at T-3 seconds, and solid rocket booster (SRB) ignition at T-0 seconds. With the Shuttle, the SRBs were required to generate sufficient thrust for liftoff, but also committed the spacecraft to at least 127 seconds of powered flight until SRB flameout. It was not possible to safely jettison the SRBs while they were providing a significant amount of thrust, and to try would likely have resulted in a loss of the vehicle (and crew).
Note that there is a huge number of things that happen both before and after liftoff. The mission doesn't begin at T minus zero; one can even argue that the mission begins years before any launch date has been picked, and the pre-launch countdown is quite specific long before the launch day.
The exact event defining T-0 is somewhat arbitrary, and different spacecraft operators can easily pick a different event to define it. Besides liftoff, I can certainly see value in defining T-0 as, for example, disconnection of some specific launch tower umbilical or, in the case of manned spacecraft, sealing of the crew access hatch. In reality, T-0 is a point of synchronization, and it's advantageous to keep it very close to liftoff as well as easy to define to a high degree of precision, but it is not really much more than that.
The awkward pause is mostly due to the commentator. He counts along with an on-screen counter until T-0. Then he waits until he sees the rocket move, or he hears ground control confirm liftoff, before he says 'liftoff'. Generally, at T-0, the hold-down clamps are released and the rocket begins moving, but it may be a few seconds before that's visible to the naked eye.
There have been rockets where liftoff occurs a few seconds after T-0, though. And others where T-0 is the planned start time, but the rocket's control system is free to delay liftoff until all the engines are running at nominal values.
On the Saturn V, if all engines are operative at Max Thrust at T=0, the Lock Down Arms detach from the Rocket Body in the First Stage of the "Soft Release". When this is confirmed on the ground, an engineer in the Control Room yells "Launch Commit!". A second or so later, When positive movement off the pad is established, the Commentator calls "Liftoff".
Indeed as the other answers point out, what happens at T0 for different series of launches or by different launch providers can vary substantially.
For Blue Origin's New Shepard, ignition of the single BE-3 engine seems to happen just before T0, and according to the question Do Blue Origin's BE-3 engines need to run for 7 seconds to “warm up”? and the Blue Origin YouTube video Replay of New Shepard Mission 8 Livestream linked there, the engine ramps in power until at least T+ 00:05 with the first vertical movement (and presumably release of some clamps) waiting until about T+ 00:07.
