Neither insuring it, nor including it on an asset list actually defines ownership. Both are equally irrelevant.
OWNERSHIP, title to property. The right by which a thing belongs to some one in particular, to the exclusion of all other persons.
You can put your name on it, you can insure it, you can add it to an asset list - none of that changes the title and right of actual ownership.
If the PC doesn’t own ‘it’, they shouldn’t be insuring it, they shouldn’t include it on an asset register and they shouldn’t assume or accept any liability for it. Unless any such liability has been agreed and assumed after agreement with the owner.
Let’s not over complicate things.
If you don’t own it, you shouldn’t ‘list’ it on an asset register, you shouldn’t insure it and you shouldn’t accept any liability for it.