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: