The rules state:

If you find a scroll, you can try to figure out what spell it contains. If the spell is a common spell from your spell list or a spell you know, you can spend a single Recall Knowledge action and automatically succeed at identifying the scroll’s spell. If it’s not, you must use Identify Magic (page 238) to learn what spell the scroll holds.

And Identify Magic says:

Once you discover that an item, location, or ongoing effect is magical, you can spend 10 minutes to try to identify the particulars of its magic. If your attempt is interrupted, you must start over. The GM sets the DC for your check. Cursed magic or esoteric subjects usually have higher DCs or might even be impossible to identify using this activity alone. Heightening a spell doesn’t increase the DC to identify it.

Critical Success You learn all the attributes of the magic, including its name (for an effect), what it does, any means of activating it (for an item or location), and whether it is cursed.
Success For an item or location, you get a sense of what it does and learn any means of activating it. For an ongoing effect (such as a spell with a duration), you learn the effect’s name and what it does. You can’t try again in hopes of getting a critical success.
Failure You fail to identify the magic and can’t try again for 1 day.
Critical Failure You misidentify the magic as something else of the GM’s choice.

It makes sense to me that the DC should be the spell rank DC. What’s not clear to me is how, if at all, a success vs critical success should be played. On a non-critical success, would they get a sense of the type of effect of the magic, but not know the specific spell?