Judges are chiefly concerned with questions of law, so unless the laws treat these two spells differently, a judge likely would not care.
Detectives are in the business of discovering facts, and so a detective would make it their business to learn (or develop) a variety of procedures to distinguish between the two spells.
You describe two differences that can be observed:
- the spells have different durations
- the movement constraints are different
The Immovable spell expires on its own, while the Ownership spell does not. So, if a "frozen" person or object is discovered, one approach might be to attempt to move the object at regular intervals, until it can be moved. If Immovable has a known duration, e.g. it always lasts 24 hours, then a frozen object that stays frozen longer than 24 hours is definitely frozen by Ownership. No conclusion can be drawn if it becomes moveable in less than the known duration -- that would be equally consistent with Immovable expiring or Ownership being deliberately released.
Immovable spell can also be dispelled by a third-party, unlike Ownership. So, if the situation permits, the detective can attempt to dispel it. This attempt would need to be performed covertly, to prevent the possibility that a sneaky caster is observing the attempt and choosing to release Ownership at the same time to mislead investigators. Or, if the dispel ritual is too overt to be concealed, the detective might try the opposite approach: perform a sham version of the dispel ritual, and see whether the object suddenly becomes unfrozen; if it does, the spell was probably released by the caster, meaning it was Ownership. (Unless the timing happened to coincide with the natural expiration of Immovable.)
Immovable freezes the target in relation to the caster, which means that forcibly moving the caster will cause the object to move similarly, whereas Ownership doesn't establish any kind of "sympathetic" movement. So, a detective might try to move every person at the scene who is a potential caster of Immovable. If the spell requires line-of-sight or proximity, that will narrow the set of people who need to be either asked to move (or manhandled, if uncooperative). Any person who refuses to cooperate instantly becomes a suspect, just like people become suspects in real-world murder investigations if they refuse to provide an alibi when asked. If it turns out that the frozen object moves when a person moves, then the detective knows both that the spell is Ownership and also who the caster is.
All of these ideas only work if the mystery spell is still active when the detective discovers the target of the spell. I think it will be next to impossible to determine which spell is the culprit if the target has already become movable when the detective arrives. This suggests that criminals who use these spells in the commission of crimes have a real incentive to bring an end to these spells before investigators arrive on the scene; in the case of Immovable, that may require the help of an accomplice to dispel it as soon as possible.
A judge would probably care less about which spell it is, and more about whether the detective did anything improper to study the frozen object, such as manhandling a person without sufficient cause who is suspected of casting Ownership. In a case like that, a judge might rule that anything is inadmissable that was learned by the detective as a result of the improper investigation.