How do I know if my pets are playing or fighting?

I have a 9 year old medium sized dog and a 1 year old cat. My dog doesnt do very well socially, we got him at 6 months old and while we think he may have been abused before he had us, i also wonder if maybe we just didnt know that a dog so young needed to be socialized. he gets anxious around other dogs, he likes to be by himself, and the few times we’ve taken him to a dog park, it seems like he doesnt really know how to play

my cat has been with us since she was 8 weeks. shes constantly jumping of the walls. when we first got her, the dog was curious but she was very scared of him and reacted aggressively. it’s mostly my fault, i could have done better to socialize them, i mostly separated them with a baby gate and then always supervised them when she figured out how to get over it

since then shes warmed up to him and often tries to play with him, but im not sure if hes reciprocating. she’ll swipe at him and he’ll run away or vocalize a bit. i get that hes nervous and i try to separate them when he clearly is trying to get away, but when he tries to get back at her, i cant tell if hes angry or trying to play back. he hasnt bitten her before, i dont think he has a high prey drive and hes generally pretty lazy, but he doest get pretty anxious

my family is worried that he’ll hurt her, and i dont want to be that guy thats like “theyre just playing!” as hes ripping her apart or something, but i also dont want to punish them for trying to have fun. i know more about cat body language than i do dog, so i think i know shes not being mean, but i dont want her to invade my dogs boundaries just because i cant read them. apart from playing, they can coexist with each other pretty well