Isn't this the old meaning of "put your money where your mouth is"? What other meaning does it have?

I was trying to think of a more clever twist on it, like "put your mind where your money is."

I always interpreted it roughly as "Do what you say/preach".

That's roughly correct, but it means that because it is saying that if you really believe something you should be willing to back up that belief with something that will cost you if you're wrong - i.e., a genuine belief that you honestly think is correct is something you should be willing to bet on because you will expect to win the bet.

People telling someone else to 'put their money where their mouth is' are essentially asking the someone else to prove the sincerity of their belief - or the person saying it has an opposing sincere belief and they would like to make some money off of a bet over who is right.