Check out C.S. Osborne NO.W-1 Hand press machine. Works great for buttons and grommets.

A button press like Eric mentioned is pretty much the standard, but I don't think the button mold that you are trying to use will work. If you only have a few to cover you may want to go to your local upholstery shop and have them cover them for you. You can get a fair amount of $$ invested in making buttons. I know I've covered several for people and don't normally charge.

The Handy button dies are marked for Light, Med, and Heavy.

For the Buttons I make the leather thinner.

Don't waste your time trying without a machine. As Hal said... if it's just a few buttons, go with the leather to a saddler who has a machine, or buy one. Anything else will only get you in a bad mood and with the quality of the Osborne you will certainly do nothing wrong. But meanwhile I also find good qualities in China at much lower prices. I have an old one myself, where I don't know the manufacturer. She doesn't work with a lever but with a turning spindle. This works very well and my tool supplier also has this type in their program. Unfortunately he is currently revising his website, so I can't post a link.
But also with the button machine you can get problems if the material is too thick. We already had the topic about making buttons some time ago.