"put down" ≠ make a down payment?
Put down does have a meaning write down which could easily extend to nominate or register in this case. See definition 29 at Dictionary.com:
29. put down
a. to write down; register; record.
b. to enter in a list, as of subscribers or contributors: Put me down for a $10 donation.
What you don't want is another standard sense — which could be applied to humans, I suppose:
k. to put (an animal) to death; put away.