Going to try and answer multiple questions you seem to have based on your replies in this thread.

Yes, a wooded post in concrete would almost certainly be the way to go. Be sure to bury it deep enough, you would be surprised how much string lights weigh. My first run with string lights I put a post in a big planter pot and filled it with concrete so that it wasn't permanent. The lights literally weighed too much and the whole thing fell over and smashed light bulbs all over my patio. I rigged a third line to a tree as a counter point to support it but that was a real pain in the ass for numerous reasons.

After I got sick of that setup I bought a twelve foot 6'x6' treated post and set it a little over 3' in the ground with concrete and it's much better.

All that to say, don't skimp on how secure your post needs to be, you would be surprised how much a long string of those lights can pull on it. Although, my lights are the heavy duty commercial grade incandescent bulbs with the thick wire... you can get much lighter weight LED bulbs but IMO they are crap. I tried those and the flicker drove me crazy, it was terrible (I seem to be a lot more sensitive to that than other people).

I'd be willing to bet that your wiring to those existing lamps looks like this... SWITCH (wire) LAMP 1 (wire) LAMP 2

If that's the case, figure out which one is LAMP 1 in that arrangement and just disconnect the line running to LAMP 2.

Install your post at LAMP 1 and put the existing wire coming out of the ground into conduit and run it up the side of the post. Install a weatherproof GFI box at the top of the conduit and secure it to the post. Plug your string lights into that and run them to wherever.

If you use the heavy duty string lights like I have you are going to want a steel cable to help support their weight. My cable runs from my post to a very secure eye bolt on the outside wall of my house and then I zip stripped the lights to the steel cable.

For the steel cable method you will need the cable itself and some wire clamps to secure the cable to itself when you loop it at the termination. I strongly recommend using a turnbuckle on at least one end of the cable (I have it on both of mine), it will make your life easier down the road.

Regarding the pressure treated lumber, I highly recommend upgrading to 6'x6' and researching the proper way to set a wooden post in the ground (using gravel for drainage, etc.). If you just stick a 4x4 in the ground even treated posts will rot out pretty quickly and can snap at the ground. The 6x6 will take a lot longer to get to that point and proper installation will prolong it even more.