What type of adaptor do I need to connect a female 3/8 OD faucet to this shutoff valve?

Question

I recently bought a new faucet for the kitchen sink and found after removing the old faucet that the new connector won't fit on the water shutoff valve:

I stumbled upon this answer and proceeded to Home Depot to buy a similar fitting (3/8" OD to 3/8" MIP).

In my rush I forgot to bring the old braided supply line to compare the male connection it used to connect to the shutoff valve to the new 3/8" MIP to see if the threads matched. Here's the comparison of the end of the supply line that was connected to the shutoff valve and the threads on the adapter I purchased. It seems like the outer diameter is very close to the 3/8" MIP, but the threads look identical to the 3/8" OD (which are smaller).

I should also mention that I screwed the adapter very lightly into the shutoff valve to check the fit and it seems to be the right diameter, but only went part of a turn due to the threads not matching, and I didn't force it any further.

I'm at my wit's end trying to figure out what the old supply line connector is called and to find any similar looking fittings I could use as an adapter for my new faucet. I also don't believe there was any o-ring or gasket present on the old line-shutoff connection. There's some sort of white residue which I assume to be remnants of some PTFE tape or something.

How should I go about connecting my new faucet to the existing shutoff valve? If better pictures are required, please let me know and I will edit my question accordingly.

EDIT: I realize this might be a bit of an X/Y problem as another solution is to replace the quarter-turn shutoff valve with a different one that has a 3/8 male compression connection on the other side. If so, what type of thread would I need from the valve to the water supply? Am I correct in guessing based on the following image that it's also a 3/8 O.D. compression? Here's a better picture of the valve:

Answer 1

As per @gbronner's comment, I removed the shutoff valve and headed back to Home Depot to see if I could find some solution.

I ended up buying this retro fit valve in 3/8" and just replacing the old shutoff valve. I couldn't find any adapters that fit the old valve, and I still haven't figured out what sort of connection would be needed if I wanted to reuse the old valve.

The original valve's female connection to the PEX supply line looked in really rough shape with the rubber gasket in bad condition. It looks like someone put some PTFE tape onto the compression thread to try to seal it, so probably good that I replaced that one.

P.S. I'm not sure if I should accept this answer given that I would still like an answer to my original question, despite the fact I've found a solution to my problem.