The idea of an individual "timeout" was one observed when I was a kid in Hull. We crossed fingers as described, but would call "Kings" rather than feinites (I wondered why at the time, but not hard enough to ask anyone). We called the game tig.

There were variants on the basic game such as "off-ground tig" in which being anywhere off the ground rendered you safe from being tigged, and "three bobs then run for your life" where a player was allowed to squat in mid-chase, and could not be tigged while squatting, but could only do so three times. Once the chaser (known as IT) had moved off after a different victim the squatting player was obliged to rejoin the action.

Policing of fair use of the systems was self-governing, and I don't remember there being much anarchy, I think the games had evolved pretty thoroughly.

British Bulldog (and its non-contact cousin Chinese Wall) took more setting up, and made no allowance for such timeouts. While the "chase and catch" idea still applied, I would not really see them as versions of tig, more as a precursor of rugby, but without a ball.