someone else suggested this, but it didn't seem to make sense since
even if I extend the vent pipe the water level in the vent pipe will
remain at the same level (the level of the water in the cold header,
right?)

You're right. I had a brain fart. It helps in a different situation not
related to your problem.

Basically, that option turns into raising the cold water cistern at high as
possible, so that it can cope with a higher pressure differential.

Christian.