Oh, yikes. Yeah, that's a huge red flag. They can promise you whatever they want verbally, but if it's not in a signed, written contract, they don't need to honor anything and if you complain that you were supposed to be working 20 hours a week they can point to the contract and say "but this CLEARLY states 20 hours a month."
I would push back and request that they update the contract to reflect the hours you agreed to work. But between this and the owner not liking to pay people, you might do just as well to keep looking. I'm sorry I know you worked very hard to get this job. Sometimes very small companies are more flexible on their policies and pay schedule but flat out refusing to pay? No.
They could also try and not pay you for any hours worked above and beyond that 20/week because those weren't in your contract. Who knows? This is protect yourself time, not be nice to them time. Do make sure you're logging all hours worked in both their time tracking system and on your own, and even what you did to fill that time productively. It sounds like a pain but it may come in handy if things get ugly later. Ugh, I'm sorry.