This was simply using a "marker generator" that had modulation capability. The TV would hopefully sync horizontally on the square wave. Today, a MUCH better way is to use a pattern generator and modulator, or a DVD player with a test pattern DVD.
Multiburst is fine for detecting high video frequency problems, but a square wave (like a white square in the middle of a black background or vice-versa) is better for detecting the low-frequency smears. These smears can also be observed with the long horizontal black bars in the Indianhead test pattern.
Tektronix produced the most comprehensive book on video distortions, although it contains much that mainly applies to transmission gear rather than TVs. The section on linear distortions is useful to get an understanding of what distortions are possible and how to measure them. In your case, precise measurement is not needed, but recognition of the various types could help.
https://download.tek.com/document/25W-7049-4.pdf