When the blower starts on a cold morning, I also should have mentioned it starts off slow and slowly increases as the engine warms up. But I thought the blower used to stay off entirely until the engine started to warm up but I can't remember for sure.
That's normal operation. Even in AUTO mode the blower will be on, only at slowest speed until the engine coolant temperature warms up.
(On '92-'04s this was done with a cold engine lockout switch (sensor mounted inline of heater hose), '95+ do it over the SCP databus with PCM supplying ECT temp to EATC)
I have no idea why your blower motor isn't going to FULL speed automatically :shrug:
If it goes to full speed manually the speed controller, motor, and wiring is fine - as the full speed relay gets tripped by the EATC when you push the thumbwheel up.