Is 'wheel alignment' the same thing as "balancing the tires"?

Not the same but you need both!

You have more than an alignment and balance. The alignment has nothing to do with vibration, and balance has nothing to do with vibration while the car is sitting still. There is something else as well, causing the vibration while stopped. Alignment and balance needs to be checked as well, but there is another issue.

poss bad motor or trans mounts