I'm guessing the video is from about 1962. In 1962 the rules for pocket billiards were confusing. From the 1962 BCA rule book:
page 29 under general rules for pocket billiards:
Player forfeits one point for:....
6. For striking cue ball twice on same stroke.
page 45 under the rules for 14.1 continuous:
Push Shots: A push shot made with one continuous stroke of the cue is legal, whether the cue ball is frozen to an object ball or not. Foul stroke is penalized by loss of one point. The referee is the sole judge in ruling on a push shot.
Frozen Cue Ball: When the cue ball is in contact with an object ball, player may play directly at object ball in contact with cue ball, provided that the object ball is moved and the cue ball strikes a cushion, or provided the object ball which is in contact with the cue ball is driven to a cushion. Failure to comply with this requirement is a foul. Penalty: loss of one point.
The Push Shot rule implies that even if the cue ball is 4 inches from the object ball, if you arrange your follow through to hit the cue ball a second time, the shot is legal. (Or even 10 inches for those with freak follow-thoughs.) This contradicts the general rule. Further, the Frozen Ball rule kind of implies that if you are frozen to an object ball, only one of those two balls can achieve the cushion contact requirement.
In any case, the referee called the shot fair or at least did not call a foul. I think the rules gave no guidance to the referee to help him judge such shots.
The rule was the same in the 1963 rule book, but in 1965 it was changed to:
(page 45)
Push or Shove Shots:
If in the opinion of the referee, the player has pushed or shoved the cue ball with his cue, it is a foul. Penalty: Loss of one point. For all practical purposes, a push shot is one in which the cue time remains in contact with cue ball after cue ball strikes object ball or when cue tip again contacts cue ball after cue ball strikes object ball.
This wording was borrowed from the general rules of carom billiards. The foul for double hits was still in the general rules as was the frozen ball rule.
This means that in 1965 the rules became essentially the same as the current rules. In 1967 the Push or Shove Shots rule was deleted from the text. This is unfortunate in that it left open the case where the cue ball is very close to the object ball so that there is only one tip-to-cue-ball contact but the action is the same as a double hit.
By 1992 the double hit rule has migrated to (page 38):
15. Fouls by Double Hits. It is a foul if the cue ball is struck more than once on a shot by the cue tip. [which is OK, but it then goes on to say ... ] If, in the referee's judgement, the cue ball has left initial contact with the cue tip and then is struck a second time in the course of the same stroke it shall be a foul. [...]
This means that if the referee believes the tip remains on the cue ball for several inches, a double hit on a ball an inch away is legal. In fact, the tip remains on the cue ball for only a few millimeters of travel.
In 1993, the rule was changed again, and the "chalk-width separation" guideline was inserted. I think that guideline was destructive. Or at least confusing.