A few years ago, I purchased a new G26 at a local business with my money and in my name. In my state, private party sales or exchanges do not require going through an FFL. At the time, she was a legal resident of my state. So, on her birthday all I really needed to do was to give her the firearm as a gift. However, just to have a paper trail (such as in the case of theft or loss from her and any subsequent investigation related to illegal use of the firearm, etc), I typed up a simple bill of sale, accepted a $1 sale price, we each signed our name, and we each kept a copy for our records.
Because you accepted money in the exchange it does not qualify as a gift per the 4473 instructions.
Question 21.a. Actual Transferee/Buyer: For purposes of this form, a person is the actual transferee/buyer if he/she is purchasing the firearm for him/herself or otherwise acquiring the firearm for him/herself. (e.g., redeeming the firearm from pawn, retrieving it from consignment, firearm raffle winner). A person is also the actual transferee/buyer if he/she is legitimately purchasing the firearm as a bona fide gift for a third party. A gift is not bona fide if another person offered or gave the person completing this form money, service(s), or item(s) of value to acquire the firearm for him/her, or if the other person is prohibited by law from receiving or possessing the firearm.
If you had just bought it and given it to her, you would have been fine. As it stands, by officially turning it into a sale and accepting money, by the letter of the law, you committed a straw purchase.
Folks, this is VERY simple.
If you want to buy a firearm as a gift, that is perfectly legal as long as the person is a resident of your state and is not a prohibited person. But a gift is a GIFT. Don't complicate things. Buy it and GIVE it to the person. Don't turn it into a sale by accepting goods or services or money in trade, don't try to get cute at the gun store by having one person fill out the form and someone else pay the money.
If you can't stand the idea of filling out the form and then giving it to the person, then give them the money as a gift and let them do what they want with the money (which could include buying a gun if they so choose). But if you do that, do the gift at home--don't be playing around with handing over money at the gun store. The FFL is not going to allow the sale if things start to look hinky.
Another option if it's critical to you that there be some kind of paper trail to the recipient is to buy the gun yourself and fill out the 4473 as the actual buyer. Then, when you give it as a gift, go to an FFL, pay them to do a transfer to the recipient and then have the recipient fill out the form to accept the transfer. To make sure there's absolutely no misunderstanding, do NOT accept any money from the recipient.