Wait is a great command. Wait to get out of the car Wait before crossing the road Wait before playing a game Wait if they are walking ahead of you and you want them to wait for you to catch up Definitely need a wait if doing dog sports. Wait to put on a lead Wait to have their feet dried Wait to hold a position at the vets - (stealth boast one of my dogs does not need sedation for x rays due to having the best wait ever!) etc
Wait for me is in any position so maybe a sit, could be a down, stand, paw up, lying on their backs etc basically it means wait just as you are. Like a freeze position.
If you use a wait I would always have a release command.
If you use a crate wait is easy to train say the word wait and go to open the crate door if the dog moves forward gently close the door. If they dog stays still give a release command and let the dog out.
Another way to teach it is on a large step. When the dog gets to the end of the step click and treat. The dog will begin to stop when it gets to the step when this happens add the wait command. You can then move this is other places as the dog has learnt that wait means just that!
Another way to teach it is with a tuggy - get a good game playing, then move the tuggy around but ask the dog to wait so that he can not get the tuggy - then release command and he is allowed to play with the tuggy - most dogs learn this way really really quickly.
I would call it impulse control rather than patience but all dogs need to learn this for their safety and also it builds up great drive and enthusiasm in a dog.