Do I address this or do I let it go?

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Not sure if there's any point addressing this, as it happened over a year ago, or if I would be overreacting. But I'm really upset and a bit angry too at this lie.

Before I met DH. His parents had separated and his dad decided to leave the house and fly to his home country for eight months.

I met DH and then a couple of years later, because of COVID, he invited me to move in with him and his mum. His mum agreed and said I could stay while we found a house.

Last year, both my mum and MIL came to see my baby when he was born. My MIL told my mum that I had invited myself to live with them, and that her husband was so upset about it he left the house. She basically said they had separated because of me. When it happened before I had even met any of them.

Maybe HE should move in with them & let you go, let you live a normal
lige, without this shit!

So what if you sound rude? Come on OP.

Have you got a whole respect your elders bollocks no matter how they treat you going on?

She's toxic. Will you step up when she tries to turn your kid against you? Your bloke is weak and doesn't have your back. Have the row.

Urrgggh you married a mummies boy who will always put her first - and she's a really nasty piece of work. Not a good situation to be in at all.

Stay away from the witch pls

Thank you all for all your advice

Definitely do NOT move in with her. You can't trust her and life will likely be difficult.
As others have said when she asks why refer to the lies she has told.
You can do this without being rude. Try to remain calm and factual. Let her know you feel hurt and you won't be living with her, and ask her directly to change her behaviour and stick to the facts in the future.
To be honest she could believe this version of events herself now if she found DH leaving her so traumatic and needed someone to blame rather than accept he didn't want her.