No, you're right.
But a lot of folks have wondered why a howler from Dumbledore, saying "remember my last," would affect Petunia as it did. "My last" seems to imply more than one letter was exchanged between them. Book 7 fleshes it out a bit more: she had actually corresponded with Dumbledore before, begging to be "taught" magic. It helps to explain why Petunia was affected more than perhaps she ought to have been by a seemingly cryptic message.
I think it would be pretty powerful to be reminded that if you send your nephew out of your home, you are virtually executing him. I can see a reminder like that hitting Petunia pretty hard!
Probably part of the letter that came with Harry was saying that if they took care of Harry until his 17th birthday (and the protections ended) he’d make sure they were still protected, but if they didn’t take care of Harry through then he wouldn’t take care of them afterwards. So the “remember my last” was a thinly veiled threat, reminding her that if they didn’t do their part of the bargain they were on their own starting on Harry’s 17th birthday. Petunia knows this stuff is real, know Voldy is a threat and knows he’ll capture and torture them to find Harry the first minute he can, and that he won’t care that they never liked Harry and won’t believe they kicked him out years before and haven’t heard from him since. Dumbledore played for keeps and wasn’t above saying “screw them they broke the deal”.
The howler was related to sending Harry from the house. The letter with baby Harry must have explained the "Mother's blood" protection in some way.
What I don't quite get, is what induced Petunia to agree to raise Harry in the first place? Love for her sister seems the only answer, and yet Petunia was so bitter.