Not really a press fit, but the end of the cocking rod is peened to hold the lifter in place or the lifter is staked (center punched) to the cocking rod.
When both barrels are fired the cocking rods have a lot of play in them, when cocked they will still have a little play but that does not look right. Your picture showing the lifter looks correct.
I would make sure that the screw holing the cocking plate in the for-end is tight.