How would y'all handle this? My thoughts so are:
If you successfully identify it at all, you get the rundown on how all golems work. Physical resistances which are usually only bypassed by adamantine. Mindless, so immune to mental stuff, plus all the basic construct immunity. Plus that golems are immune to most magic, but are healed by one type, hurt by another, and slowed by a third, plus usually a specific weakness by type of golem. These are just things you should know about golems, much like once you recognize that a monster is a demon you'd know that it is weak to cold iron and good because all demons are.
I also, as a general rule, read most of the descriptor text to my players and will use stuff mentioned in there as the creature's most well known feature.
However, the the specifics of the antimagic have profound tactical significance and do not seem like they should be immediately laid bare for a single normal success. I think you should get at least one piece of information specific to this creature, and hopefully one that is combat relevant. Most golems leave us with these to choose from:
I'm always hesitant to go to special attacks on any creature, because often it doesn't really change how PCs approach them. Knowing a Clay Golem's wounds will curse you doesn't really help if you still need to slug it out with the thing.
So the first four are the best choices. My thought is that for a normal success on an initial roll the players would get the "Vulnerable To." While not exploitable for every party, they are the most intuitive and therefore the most obvious thing to come to mind first.
On futher attempts to gleam additional information at a higher DC, or on a critical success, you'd get that plus one of the other 3. Not sure how to best pick which one to share, but I'd think harm would be the most helpful. Maybe roll randomly to determine it? An on subsequent rolls at even higher DCs they could learn the rest of the information.
One issue I foresee: once a character has attempted an incredibly hard DC for their level, they have exhausted everything they know on the subject. And since golems start at uncommon (hard) and can also be rare (very hard) you don't have many checks before you hit incredibly hard. I would think that someone should be able to piece together a pretty complete list of a golem's abilities and antimagic, given enough successes.