So you talked to you MIL about being kind to your DS, who has had a bad 18 months. She responded by telling you not to tell her how to show her love for her child. Her child is your DH though, not your DS, who is her grandchild. So this does not make a lot of sense, or, you got your wires crossed.
She's allowed to express her irritation with your DS's behaviour to her son. She may be doing this as a way of getting it out, without having a go at your DS, who she is doing her best to be kind to. She might also be saying, more practically, that if your DS is going to make a mess, your DH needs to clear it up.
This doesn't sound much to do with MIL / DIL relationships at all really. More to do with two families who have become deeply entrenched in their own ways of doing things over lockdown and are both struggling to adapt to others' company, in close proximity, for an extended period.
If the holiday house is MIL's, she will see you all as her guests, herself as your host and expect you to fit in with her ways of doing things. As guests you have to respect that. Her response to you is more about that than about who you are.
I'd be pretty pissed off with my guests, or anyone anywhere actually, telling me how to behave towards other people and specifically to be more touchy-feely, when that is not my style. It's incredibly disrespectful, patronising, arrogant, over-stepping behaviour on your part.
You can tell her about your DS's problems and what you are doing to be kind to him, in the hope she'll understand and follow suit. You cannot tell her how to behave. She is not your agent to direct.
So, from what you've said, I think the person getting the MIL / DIL relationship badly wrong here, as well as the host / guest one, was you.
You've been stuck in your own family bubble for far too long, it's altered your perspective and your behaviour. You need to re-learn social interaction.