For me, a down-payment is the cash you put up to secure the purchase of a house or apartment. I guess it could be money borrowed from a relative, or even a bank, but usually down-payment does not refer to money secured from a bank or other lending company.
CALD:
down payment , downpayment noun [ C ]
an amount of money that you pay at the time that you buy something, but which is only a part of the total cost of that thing. You usually pay the rest of the cost over a period of time
In the UK I believe the terms "down-payment" and "deposit" are used interchangeably when talking about property purchases.
Collateral is the bricks and mortar with which the lender secures your loan. It is his guarantee that he takes no risk in lending you the money. You don't pay, he repossesses the house.