tightening belt tensioner after fitting new timing belt

Initially any slackness in the belt will be spread between all of the pulleys. By tensioning it, then cranking by hand, you are moving all of the slack to the location of the tensioner. The tensioner can then be adjusted correctly.

Edit - You will appreciate that there is a lot of friction associated with each of the pulleys that the belt travels around. If you try to turn them by hand it is quite difficult.

There will be a very small amount of slackness in the belt between each pulley after first fitting the belt. If there were not slackness, you wouldn’t be able to get the belt on.

The initial adjustment of the tensioner will not add enough pressure to overcome the friction of the pulleys, so they won’t move and any slackness between them won’t be taken up. By rotating the engine by hand, you move the slack between each of the pulleys so that it is now at the location of the tensioner. You can then adjust the tensioner to take up the slack.

By repeated adjustment you are ensuring that any remaining slackness in the belt is taken up by the tensioner.

Another edit -

When you said in the video that he tightens it four times, I believe what you described as the first 2 times was just 1 attempt. He was just struggling to get it tightened. The very last time he was just making sure that the bolt was tight before he finished.

I do think that he didn’t do the initial tensioning correctly. He turned the Allen key anti-clockwise before locking the bolt. I believe he should have turned it clockwise to put the maximum tension on the tensioner as described in step 63, but please do not take my opinion as correct. Maybe someone else will back this up.