Do old CRT TVs and monitors distort their image, or cut it off? I vaguely understand that an old CRT monitor/TV's screen is a section of a sphere, and since the section lines of a sphere curved, the edges of the screen are curved as well. 



My question is, if you take a 4:3 image and one of these CRTs, is that image being stretched and 'fish-eyed' to fit the curved edges of this monitors, or does it appear square, but with its corners cut off?

In other words, is A or B correct? Note that the blue lines are straight. On A the image is square and orthogonal, and on B the image is warped to conform to the borders of the screen.
CRTs were originally round, as can be seen in this 1945 prototype. The cabinet masks off the outer parts of the tube to create the square viewing area. This is called overscan. Later tubes were more rectangular, but still relied on overscan. This was partly because as the tube aged, the scanned area shrank, so the cabinet would hide the blank parts of the screen. The front to back curvature of the tube is not accounted for in the image because every model tube had a different curvature and I don't think it added much distortion.

Curves at the rounded corners of the cabinet were also not accounted for because of the differences in cabinet. But that was part of the motivation for overscanning and for limiting important picture elements to safe areas.