advice about sink traps

Well, as I'm sure you're aware, a trap is basically a U bend which holds (i.e. "traps") some of the water you flush down the sink. This is is so that bad smells from the sewage system can not get back into the room.

The different types simply have the outgoing waste pipe presenting at different angles.

So, with a P trap, the waste pipe exits horizontally. A bottle trap also has horizontal exit, but takes up less space.

With an S trap, the waste pipe exits vertically. A slimline pedestal trap also has vertical exit, but takes up less space.

Anti-syphon and anti-vacuum are (as far as I know), different terms for the same thing.

Normally, the pressure in the sewage system is the same as atmoshpheric pressure. This means that if you had (say) a conventional P trap made out of clear plastic, you would notice that the water levels either side of the U bend are equal.

However, under certain circumstances, the pressure in the sewage system can become reduced which means that the residual water in the U bend gets sucked down the drain and the smell-sealing effect stops, leading to a nasty niff in the room.

AIUI, Anti-syphon bottle traps have clever bits inside that allow the pressure to equalise without losing the residual water in the U bend (which isn't actually a U bend in a bottle trap, but does the same job).