First of all, don't worry, and don't doubt yourself.
I think what you are experiencing is the classic struggle when the pecking order of the matriarch gets shaken up.
It is pretty common for mothers of new parents to feel displaced if their DD or DDIL has just given birth. They are no longer 'the Mother' in the family. i.e. the mother who is doing the most hands on day-to-day care of a child. A lot of mothers accept this, and take it in their stride, but some kick against it, and I reckon that's what is happening here.
She is trying to make you seem useless, or at least, not as good as her. She is doing it in a jokey way, but, make no mistake, there is a negative intent behind it, even if she does like you.
It's not really personal - she would probably be like this with anyone her son had a DS with. And the most tragic thing is, is that she is fighting a losing battle. First of all, it does not really matter who 'The Mother' is in the family, and secondly, she can never take your place in you DCs eyes, or in the eyes of your DP who you had the child with.
Basically, she is trying to start a pissing competition that she can't win, and on some level she knows this.
So my advice? Ignore her when she is rude. She will probably get worse before she gets better, and so if your DP is aware, he may pick her up on things. If you tackle her you'll get a tinkly little laugh, and be told not to be so sensitive.
And remember, she is lashing out because she is the one who feels ousted, and is sad to not be in your position.
Good luck.