Some movies were shot in 4:3 (ish) and cropped in the theater. Titanic is an example.

(For the pedantic, yes there is a name for this technique, and yes the ratios aren't exactly 4:3.)

35mm film is more square than rectangle. When you shoot wider aspect ratios, the whole frame is exposed. The eyepiece on the camera has lines on them to allow the DP to see the framing for the desired aspect ratio vs the whole frame. So it wasn't just a Titanic as example, it was pretty much all film was shot like that.

When digital cameras like the ones from Red came out, you can tell it the aspect ratio so it only saves the active pixels of the full sensor and ignore all of the out of aspect pixels. That's a brave operator doing that, and I've only seen it in the wild once.

Not quite but close. Titanic, like a lot of movies of the 90s and 2000s, was shot in Super 35. In Super 35, the image is ~4:3, but it requires optical printing or scanning to produce a release print, since the image is both not the correct aspect ratio and also occupies the area used for optical sound.

So it was not "cropped in the theater"—the theater got a standard anamorphic print. To go from the Super 35 negative to the anamorphic print, they both cropped and optically squeezed the image (in the case of the non-vfx shots), and scanned, cropped, squeezed digitally and printed back to film (in the case of the vfx shots). This was a few years before they did full "digital intermediates."

The term is "open matte" and is the ideal way of watching movies on folding phones.

https://en.wikipedia.org/wiki/Open_matte

Open matte is a filming technique.

The second half of your sentence is misguided.