Should I notch deck posts

Around here, 4x4s are not uncommon. This table shows that a 4x4 pine post that is 6 ft or shorter can support a load area of up to 10 ft² assuming 50 lbs/ft².

Don't trust me or a random table on the internet; make our own calculations from tables in the building code or other trusted sources and consider having them checked by an engineer.

I really prefer post bases for decks over putting the posts in concrete; it's a much better way of going. I like the beefy ones like the EPB66HDG, but Simpson makes a lot of different ones.

Here's a trick for getting them right. Dig the footing holes, and then temporarily attach the connectors to the beam. Put the beam on the ground directly under where it is going to be, level it in both directions and anchor it (stakes or whatever) so it can't move. Then, pour the concrete around it. This gives you post connectors exactly where they need to be. After the concrete sets, just remove the beam, put the posts in, and put the beam on top of them. Simple. If the posts are all the same height you can put short pieces of post between the deck and the beam.

I like hefty footings for 2nd floor decks; I would go at least 1x1x1 (depth set based on your frost line); I have a 2nd floor deck with a roof over it that has 5 footings that are each 2'x2'x'1. You can put a bit of rebar at the bottom to help hold the footing base together.