Programming wise, I would definitely pick
But in a broader usage sense, I find myself using idiomatic terms more than programming terms anyways. In addition to the other answer's suggestions, baseball terminology sprung to mind when I read your questions.
At bat and variations
- Barry Bonds is up to bat looking to extend his home run record even further.
- Casey at the bat
As a bonus, you get the term On Deck
In baseball, on deck refers to being next in line to bat. In a professional game, the batter who is on deck traditionally waits in a location in the foul territory called the on deck circle.
Baseball is more complicated than a simple turn based game (teams take turns on offense and defense in addition to players taking turns at bat), but I think the "at bat" and "on deck" phrases extend nicely to simpler, non-team based games.
Serve Other sports-like games might use a concept of "whose serve" is it.
- Player 2 is serving.
- Team A wins the point and is now serving.
I also know you're not looking for every day usages, but I thought it would be useful for future searchers/visitors.
Terms I've seen used in other games for the player currently able to move:
- On point
- Initiative - Player 2 has the initiative: all other players can only defend.
- Edge - I first remember this from someone teaching me about Vampire: the Eternal Struggle (also known as Jyhad), a collectible card game from Wizards of the Coast (I played Magic: the Gathering quite a bit...). However, the concept of "The Edge" is quite a bit more complicated in that game than just 'whose turn'.
The nice thing about noun-like names for the turn concept is that you can apply verbs and modifiers to them to describe non-standard actions, or just describe them in different ways.
- Player 3 interrupts, seizing the initiative! Player 2's turn is over.
- Player 4 now has the edge.
Terms like these are also better at conveying the concept for games that allow certain actions to be taken when it's not your turn. If your game is especially complex, there are a number of planning, investigation, reading, responding etc. type tasks that the inactive players could take when it isn't their turn. This can greatly speed up the game as well as keeping the non-active players more interested instead of just waiting.
Since you're specifically talking about card games and you mentioned poker and in/out players in the comments, you might find some useful suggestions in this wiki article on betting in poker.
- Under the gun - the article says this refers to the player making the first bet in a round, but I see potential for expanding it.
- Betting player - if your game engine is specific to betting games, this might be the best choice. If you are trying to leave it generic, for concepts which extend to different games or types of games, a more generic
executing player is probably better.
And as I mentioned in a comment, you might want to check out http://gamedev.stackexchange.com for more input on the game-specific side of your design.