We don't actually know the content of the note, the OP didn't post it word for word. And in her first post she doesn't call it confrontational or extreme or anything else.
The OP also admits that her own hormones may have made her own reaction to it irrational, so perhaps what she does say about the note's content is coloured slightly by her own interpretation of it.
And the OP's husband thought her response was wrong and stopped her from retaliating with a "offended and grumpy" reply, complete with some earplugs she just happened to have handy, so he can't have thought the note or the neighbour was in the wrong.
Plus the OP has said she has already apologised to certain neighbours but not others and has accepted that she does at least owe this particular neighbour an explanation.
We don't know one single thing about this neighbour, we don't know if she is ill or has a difficult job, if she is going through a hard time for some reason, if she has covered her own house in egg box insulation and tried earplugs before she complained or if she is just a moaning cow who hates babies. Or for all we know she has lost a baby herself and finds it hard to listen to someone else's child cry. We don't know. So it's not fair to judge her on the basis of her note, as even the OP believes she herself overreacted to it.
All we do know that the OP says the neighbour has kept quiet for weeks, without complaining, and has seen the problem get worse with no explanation for it. She hasn't complained at the first bit of a noise, she's complained after weeks and weeks of escalating noise through out the night. And as we don't know exactly what the note said, we don't know if it's tone was desperate rather than angry or even jokey rather than aggressive or rude.
As someone said up-thread (think it was KerryMumbles), if you know you are causing your neighbour distress and disturbance, even if it is something as natural as a crying baby (toddler in the case of the OP's child, not a newborn) then you should at the very least apologise and try to do something to stop that disturbance.
And in this case, when the noise is a crying child, it's not unreasonable for the suggestions to include the OP trying to insulate the wall in some way, move the baby away from the party wall or look into the possibility of medical help to stop her baby from suffering so much pain. It's not like anyone has suggested she gag her child, they've suggested wall hangings and changing bedrooms and a quick trip to the doctor to see if the OP's child might have an ear infection rather than teething problems.
Which isn't a bad idea considering I thought my LO was teething this week and it has turned out to be a throat infection he is now taking medication for. I took him to the doctor today, following two disturbed nights, nobody laughed me out of the surgery and we just might have caught his infection before his tonsils turned septic (as the doctor so nicely put it). And yes, this morning I did ask my neighbour if we disturbed him and he said no.