My Tuppence : OP needs a passport.
The answers are interesting: the etymology is without contest. Its use in AU/NZ - and probably by the chap in the origibal post - is the same as in the UK.
It is to take a calculated risk for gain, often from a relatively disadvantaged position - the very nature of risk. In the rugby-loving AU and NZ, punting is definitely a "rugger" term in current usage, but stated with its roots intact every time it is used. Rugby lovers know that a punt is a risky kick, with the potential for massive gain, but with the risk of likely loss of possession.
In American football, punting is derived directly from Rugby football, in which a punt is one of three types kick (punt, drop and place) in which the ball never touches the ground after leaving the ball carrier's hands.
It is often a get out of trouble strategy to move the ball forward becuase it is near the goal line. For this reason, punted kicks in rugby, taken from within the 25 yard line, are almost always diagonally aimed downfield, to attempt to cross the touch line within the opposing side's 25 yard line. That's a large "yardage" gain acquired from a former position of risk (within one's own 25 yard line). The side which punted has moved the ball all the way downfield, at risk of loss of possession, to increase the chance of scoring and mitigate risk of being scored against.
The punt is a gamble of a kick, taken to offset risk of a try (touchdown) by the opposing side. In US football, the punt is typically taken for the same reasons: when the team is on near their own goal line, at the 4th down (nearing loss of possession (offense), the purpose of the punt is to get the line of scrimmage all the way down the field and mitigate risk of scoring by the opposing side. In both games, it is a likely loss of possession.
The physical mechanism of a punt kick: the ball is let loose in the air in from of the kicker, and she or he kicks it while it is aloft, without it touching the ground, usually at a diagonal, with the aim os a singe bounce then out of touch far downfield.
This technique differentiates the "punt" from a "drop kick" because the drop kick is allowed to bounce and struck immediately after touching earth, with the purpose of sending the drop-kick ball between the uprights of the goal posts, and thereby gaining 3 points. One cannot gain points by punting between the posts. Only place-kick after try, or drop-kick, on the fly.
In both games, a punt is a gamble... hence the classical derivation from gambling against the is correct.
In rugby (and US football) the punted kick is risky because it leads to probable loss of possession of the ball, by kicking it forward, downfield, over the opposing sides heads, in the hope of regaining it, or getting a "line out" downfield, and thereby gaining a massive kicked advantage.
The risk: if the ball is caught or recovered by the opposing side, before going out of touch, the punt has caused loss of possession.
The full back often takes a punt after catching the ball near the goal line, which is the end zone in American football. The punt is used to loft the ball over the opposing sides and is chased by the kicker to avoid an offside status (that derived (as is rugby football) from proper Football (known as "soccer" in the US).
It is usually taken while running and often by the Fullback after catching the ball inside one's 25 yard line. A punt "return" is a kick made from inside the 25 yard line, often by the Fullback, after catching an incoming kick. Punting from inside the 25 yard line which will stop play where the ball goes "out of touch" - and to result in a "line-out" (from which "line of scrimmage" is derived) at that yard-point and side on the touch line. A punt taken outside one's 25 yard line must bounce (touch ground) before going out of touch to achieve the same forward yard gain. If it goes out of touch without bouncing, then the line-out occurs at the yard and side of putch (field) where the punt was taken.
Punt is also a small angular nosed boat, as noted, and the "pont" gun was usually strapped to a bench in that boat used for killing larger number of waterfowl.