If someone asks you to buy a gun, or you offer to buy a gun for someone you are acting on their behalf. My understanding is if you're "acting on behalf" of someone that's a straw purchase according to my interpretation of the BATFE's position.
I don't agree with the law (or really, the BATFE interpretation of the law), but I don't understand why there is confusion about it.
In this case, the guy's uncle sent him a check and asked him to buy a gun for him. The guy was acting "on behalf" of his uncle.
If someone asks you to go into the store where you work and buy a TV for them because you get a discount and you do that, you are a straw purchaser of that TV. Even worse for this pair of guys is that they documented it beforehand by the favor-asker writing a check for it before the purchase.
You can buy a gun with the purpose of reselling it. Or giving it to the police (Mark Kelly's AR 15 deal). Or gifting it. It's not a straw purchase. If you have an agreement with someone to get the gun for them, then it is. It seems pretty simple to me.
Unless I'm wrong about it! (It happens...)