Am I OK to use the “bigger” size lockring (the one which uses a BB tool) to put a rotor on a 12 mm hub?

Or do I need to use the “smaller” size one (the one which uses a cassette tool)?

It fits. To me the difference only seems to be how it’s machined for the tool. I just don’t know if anything else would matter…

I think you’re fine* – more the other way round IME, where the little rings can’t be done up if there’s a biggish axle in the way.
*Wait – no you’re not.  Then we’d end up having only 1 standard size and that’d NEVER do

Your * is the reason I am checking if I’m missing something @scaredypants

Depends on the lockring? They’re not universal between manufacturers.

Which is a PITA as it means centerlock to 6 bolt addapters aren’t universal either.

@savoyad – I was kidding.  In my experience they all have fit on any hub, just that the narrow ones can’t be done up “over” a bigger axle.

Course, I haven’t tried all hubs or all rings but if yours tightens on and has a decent amount of thread engaged, then I can’t see how it could fail

What make of hub/ring are you wanting ?  Maybe somebody’ll know for sure

The lock ring says Shimano on it (but might not really be shimano.)
The hub is on an OEM Scott/Syncro RP2 (but is really Formula).

My instinct was the same as yours. I’ll just try it.

My experience was that there are several locking types.

Cassette type – can be screwed into place, but not tightened, if the axle takes up too much of the hole in the locking, as the tool won’t fit.

BB type – you may need the spanner type of BB tool, rather than the socket type, if the axle protrudes far enough to stop the socket being seated correctly.

Thin BB type – protrudes from the surface of the rotor by less, and is required if, like my bike, there’s not a lot of clearance between the rotor and the fork.

The thin BB type only seem to be available as part of the 6 bolt to centrelock adapter

https://www.tredz.co.uk/.Shimano-6-Bolt-Rotor-To-Centre-Lock-Hub-Disc-Adapter-SMRTAD05_87930.htm?sku=299902&utm_source=google&utm_medium=cpc&utm_campaign=google_shopping&gclid=CjwKCAiAz–OBhBIEiwAG1rIOu3TZC3rstvsoJe8-2LCOaNxOkuIMU-gZQwUW0mArG-hq42QI5QXJRoCzSQQAvD_BwE

As long as the BB style clears you’re fine. I’ve seen some tight tolerance rear ends. My Spark for example wont fit some brake calipers with the 180mm mount and iirc was really close to interference with a BB style lockring.
There are also some brake rotors that don’t work with bb style and 6 bolt adapters though but still work with the cassette style.