4 out of 6 not true. Carseats and gate-checked strollers *are* not counted as carryons by US domestic airlines, but the rest certainly are counted. You are allowed one carryon (as in a 22" rollaboard size maximum) + one personal item (no larger than a briefcase) per ticketed passenger. 2 containers is the maximum limit, though if you can consolidate and stay within the size limit, that's fine. The idea is that you are allowed to put the larger bag in the overhead bin, but the smaller one must be able to fit completely under the seat in front of you.
Lap babies are not ticketed, and they do not get a luggage allowance at all. In the case of a lap baby, the diaper bag is the adult's "personal item" -- well, that is presuming that the diaper bag is small enough to fit under a seat. If it is a really huge diaper bag, it will use up your carryon allowance. If Mom needs a separate purse in that situation, she had better make sure it is small enough to fit INSIDE the diaper bag.