I insert the cylinder on the left.
Starting with the hammer in the down position, after I barely begin to cock the hammer, the cylinder will slip right in. Once it's in, I bring the hammer to 1/2 cock which holds the cylinder fairly centered and in place for the base pin to be inserted.
Then I'll usually push it out through the right side.
If I botch the insertion by cocking it too quickly for the cylinder to slip right in, I'll just fully decock and start over again.
The key for me is to start to barely cock the hammer simultaneously with inserting the cylinder.
So there appears to be different methods that work, and mine is a Pietta.