The dominate spell specifies
You take command of the target, forcing it to obey your orders. If you issue an obviously self-destructive order, the target doesn't act until you issue a new order. The effect depends on its Will save.
Critical Success The target is unaffected.
Success The target is stunned 1 as it fights off your commands.
Failure You control the target. It gains the controlled condition, but it can attempt a Will save at the end of each of its turns. On a success, the spell ends.
Critical Failure As a failure, but the target receives a new save only if you give it a new order that is against its nature, such as killing its allies.
The Controlled condition is pretty straightforward
Someone else is making your decisions for you, usually because you're being commanded or magically dominated. The controller dictates how you act and can make you use any of your actions, including attacks, reactions, or even Delay. The controller usually does not have to spend their own actions when controlling you.
I'm guessing the critical failure caveat is a fail-safe to allow a second Save because you do not have access to the repeated Saves granted by Failure, but the text almost reads (as it was in Pathfinder 1e and similar RPG's) that the extra Save against attacking allies was already implied. Mostly curious if I missed any text that would indicate the second (or a different) interpretation.
Best Answer
No
Sans the context of other RPGs (especially Pathfinder 1), there's no reason to read the Critical Failure effect into the Failure. If effects are shared between different degrees of success, they are explicitly specified - see, e.g., Terrifying Ammunition. The critical failure reads:
Since Dominate lacks any similar language, there is no reason to port over effects from the Critical Failure to the Failure. I agree with your intuition that this is to give a "way out" of Domination when you Critically Fail.
In Pathfinder 1, since Dominate Person worked all duration without free repeated saves (unless you failed to concentrate), the "forced to take actions against its nature" clause was necessary to give some chance of the dominated creature becoming un-dominated. Given that the Failure effects of Dominate in PF2e provide this in the form of saving throws every round, there's just no real need for a similar clause until you get to Critical Failure.