Nowadays "aim assist" is just the name given to the whole suite of controller aim assistances. At least at the studios I worked at. It includes:

• Snap : "snapping" the aim to the closest target. For an easy example, it's typically very strong in Rockstar games, where you trigger it by simply pressing the aim button
• Tracking : automatically tracks moving targets, if the player is aiming at them (again with a Rockstar games example: this is what happens after the "snap" when you just hold the aim button)
• Head/weakpoint assist : reduces sensitivity when aiming at/around heads (or other weak spots).

These terms are just the ones I use personally, but at least none of my colleagues seemed to think they were too vague or misleading. There are also a lot of other mechanisms but these are the main ones. But I think you're right in saying that the dramatic snapping assist most games have nowadays have made the head assist mechanism almost redundant (though it is still present in many games, where it can come in handy for minute adjustments after the snap)