In particular, is it OK to use no article (to avoid confusion with the idiom "get in the game")? Are there any better ways to tell someone to log into the game without saying "log in"? Example: "get in game tonight to claim your rewards".
get in game
You're misunderstanding the phrase. The correct parsing is:
Where "ingame" refers to:
1) (adj) The state of being inside of a full screen and online game.
I am aware that UrbanDictionary isn't the most reputable of sources, but this word is very informal and relatively new, I couldn't find a better reference.
This phrasing is very idiomatic in English (unlike the word "ingame" itself):
I want to get drunk.
I want to get lucky.
I want you to get busy.
I want you to get ingame now, we need a healer to join our party!
This also means that the premise of your question is flawed. "get in the game" and "get ingame" are not directly related, this is not an omission of the definite article.