The real answer is to talk to the bank.
In the case of the last car loan I got, the answer is "no". When I asked them about rates, they gave me a printed sheet that listed the loan rates they offered based on how old the car was, period. I forget the exact numbers but it was like: New car: 4%, 1 year old: 4.5%, 2-3 years old 5%, etc.
I suspect that at most banks these days, it's not up to the loan officer to come up with what he considers reasonable terms for a loan based on whatever factors you may bring up and he agrees are relevant. The bank is going to have a set policy, under these conditions, this is the rate, and that's what you get. So if the bank includes the size of the down payment in their calculations, then yes, it will be relevant. If they don't, than it won't. The thing to do would be to ask your bank.
If you're only borrowing $2000, and you've managed to save up $11,000, I'd guess you can pay off the $2,000 pretty quickly. So as Keshlam says, the interest rate probably isn't all that important. If you can pay it off in a year, then the difference between 5% and 1% is only $80. If you're buying a $13,000 car, I can't imagine you're going to agonize over $80.
BTW I've bought two cars in the last few years with about half the cost in cash and putting the rest on my credit card. (One for me and one for my daughter.) Then I paid off the credit card in a couple of months. Sure, the interest rate on a credit card is much higher than a car loan, but as it was only for a few months, it made very little real difference, and it took zero effort to arrange the loan and gave me total flexibility in the repayment schedule. Credit card companies often offer convenience checks where you pay like 3% or so transaction fee and then 0% interest for a year or more, so it would just cost the 3% up front fee.