I need to replace some bent (30 year old) hinges on our kitchen cabinets, but I don’t know what to search for. I have looked in the big box stores and haven’t seen any that resemble the kind I need. I’m hoping that I can order some online, I just don’t know what search terms to use.
I think that's a partial wrap overlay hinge. Overlay means the door sits on the front of the faceframe when closed. Partial wrap means the cabinet side of the hinge wraps along the edge of the faceframe where it is attached. There are more components in the hinge though - maybe it's self-closing?
You will most likely not find an exact replacement for those hinges. The face frame and door looks like they even have cutouts to allow that hinge to fit and work the way it is intended. There are hinges that will work in its place, but the thing you do not have the info o is what the "overlay" is. This is the dimension that the door overlaps the frame of the cabinet. It can be 1/4", 3/8", 1/2" and 3/4". Some doors can have even larger overlay. Yours, just to guess, looks like it may be a 1/2" overlay. Here is a link to a hinge that will supposedly work in your case. It have more mounting holes and some are elongated. The extra mounting holes will get past the cutouts the original hinges have since if you tried to use a different hinge that had mounting screws in the same place the screws may not hold because of the cutouts that the original hinges used. The new hinges having screws in different places will allow the mounting screws to go into new wood.
I found what I was looking for. Shout-out to the comments and answers for pointing me in the right direction. The keywords I was missing are Self-Closing, Double Demountable, and Overlay. My particular hinges are 1/2" (13mm).