How to distort one corner of an image without moving two sides (read, hard to explain in Subject!)

BTW I continued the OP's use of old B&W movie images, and chose Casablanca as I was there a few weeks ago.  The movie was actually filmed in California, but an enterprising Moroccan man saw an opportunity and opened this cafe.

Not at all. I'm still searching for help and I can't believe that there's no way to do this in photoshop but, apparently, it's a dead end. If you find a way, I'm still searching!

I don't know if this is what you're talking about - but the only way I can get one corner to move (not 2 corners), is to make sure the layer is rasterized 🙂 

Looks exactly what I need but… I do not own the full CC, only the Photographers version (no After Effects). And, for reasons hard to explain but the main one is that images are 60000 pixels wide sometimes (before many transformations) and every time the correction has to be done only in a slice of the image, it's very impractical to switch software. Anyway, you understood my point and that's exactly what I need, something that squeezes a side without applying any perspective. Is it possible that this can't be done in Photoshop???

Unfortunately not. I've tried many times following every tutorial. Perspective warp distorts image exactly like Transform>Distort and or Perspective. In the case above (my example) heads get the horrid shift. It's ok for other kind of corrections, where you need a perspective. My case is simple geometry, no perspective involved and none must be.

Unfortunately not. The new additions to the warp tool give finer control, that's true, but give NO help if you need to keep lines straight. As said above, I don't want to curve anything, I need to compress lines from one corner without shifting the cotent towards the side I'm compressing, and need to keep the perpendicular lines (horizontal if I compress vertically) straight, creating diagonals where before I had parallels. The movie I've attached shows what I clearly don't want.