It sounds like there are some local variations but this is posted on the box controller/scorer shack at a club that has regular races. Traditional rule and that's at every bird not just your last one.

Rules vary from club to club, but if you're shooting 50 cent cartridges at a ten dollar bird, give em a dollars worth each. The ammo is the cheapest part of the shoot.
I can't tell you how many times I've seen a "dead" bird get up and get over the fence.

Back when I used to practice shooting hand thrown birds, we used a wobble trap set up on a platform so that the release height from the ground was about 7', and the minimum launch angle high enough to just clear the wire. We would always shoot "pieces" of the target for barrel number two, even if it was just a puff of grey dust.

When I started shooting targets, everything was practice for hunting. When I started shooting live birds, everything else was practice for the ring. (hunting excluded)
American Doubles is great practice for developing second barrel skills. International Trap is great for learning to see and react to targets quickly.

Just remembered that I once bought a pretty mean GH 0-frame, 16-bore Damascus gun that a previous owner had chosen to eliminate the safety on thereby converting it not into a 'live bird gun' but into a gun that better suited his needs and purposes. The plate with the initial "T" in Olde English Script (or the numeral "5") could be brass or it could be gold... never had it tested. But it's certainly not a "Live Bird Gun" beyond the fact that he hunted live upland birds.

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