For us the "wait" is generally a "pause what you are doing until we say otherwise" - it may be followed up with another command (recall, retrieve or similar) or a release (ok) to carry on with what she was about to do... While, as Mariaholgate says, the stay is stay in the position I have put you in while I go away and then I will come back and release you.
Honey has both (though the stay isn't on a command yet... it's a long work in progress!!!).
For wait we started with a treat on the floor, but us holding her lead to prevent her getting it.... we would then release with an "ok" and she could get the treat... we also practised this at mealtimes, making her sit and wait until we said ok.... I can now ask her to wait on a walk while I get bits of twigs out of her fur, wait at doors etc...