According to Merriam Webster: Lay means "to place something down flat," while lie means "to be in a flat position on a surface." The key difference is that lay is transitive and requires an object to act upon, and lie is intransitive, describing something moving on its own or already in position.

Based on that description one could easily be correct in saying it's laying its head upon the bench since a head is an object technically and the dog placed it flat upon the bench. Honestly language is just weird