Yes there is a clear but definite difference between permissive parenting and gentle but firm.
You can have boundaries and be firm and rules and still try to find alternatives though can't you?
I also have friends who's child is frankly indulged and set no limits to behaviour, in that if he hits then he is comforted with no disucssion of consequences or impact. That wouldn't be my preferred route.
I think you can have comsequences, natural ones are best, the old, if they choose not to wear a coat okay their choice but a natural consequence is they'll get a bit chilly and soon be asking for one, but it was their choice and they still felt in control of their lives and you avoided a fight.
The hitting, a natural consequence is that the happy person you were playing with is now hurt and doesn't want to play. If you (ds) had done something different then everyone could still be playing.
So you can try and offer alternatives, when my dd gets frustrated with her little bro for snatching her toy I have taught little tricks to get what she wants, she can try asking him nicely, she can try distracting him or she can pretend she's interested in something else so that he wants that new thing then she can go back to the originnal toy. Or she can tell me about it. It isn't okay for her to snatch back.
My ds is only 18m so I and my dd make allowances for the fact he is still working out how to play and share with us. We say, hey Slurboy why not do x, y, z? and if he still pisses her off, I do step in and take him to play with something elase, because my dd is only 3.5 and I make allowances for the fact she can't be expected to cope with snatching all the time, and I don't want her in a position where she feels so frustrated she wants to whallop him.
I have rules, but be flexible. I have boundaries but I try to understand when they cross them.
My dd has hit but most often it is for a veryt understandable reason so I talk to her, she feels bad anyway no need for me to make it worse really, I offer alternatives.