Yes, the woman is out of line. Not because any one comments was wrong, but because she kept going on about it.
But you do realize that you repeatedly refer to your baby as fat. See bolded.
I’m not OP but different words have different connotations.
OP didn’t call her baby fat. She said her baby is chubby.
Meaning while in Merriam-Webster dictionary they have same definition, words will mean something different based on how they are generally used.
The word fat has negative connotations. Fat - think overweight adult struggling with diet or other issues causing obesity.
Chubby is generally associated with kids.
If someone were to call it baby fat it would be different. (This is just what is generally politically correct at least from what I know in my circles )