As far as the comment about having never tried deliberately presenting a target to the attacker - if so your practice or instruction is lacking. You should be doing this every time you practice a technique. You and your partner establish ma-ai, then you present the target to signal uke and present them with an attack opportunity. I see this misunderstanding all the time when people expect to receive a munetsuki with their hands in the way, or a katate dori with their arm dangling limply at their side. No. You invite the attack, you present the target.
During the normal training routine we tend to do this type of presenting automatically rather than as a conscious effort(although I too have experienced the same sort of misunderstandings generally from novices). To further clarify, I was refering to situations where nage doesn't know what type of attack he is receiving like in randori or street encounters. In our multi attack randori training we do tend to move towards one of the opponents to draw out their attack, which I suppose is presenting the whole body rather than an individual part. Hopefully I'll remember to try it out on my next randori session!.