The issue doesn't appear to be frame play at this point. The cylinder isn't mounted solid to the frame, it's suspended on the cradle (for want of a better word) by the retraction springs like most of the cheapies. It's the mostly lateral movement of the cradle as the slack is being taken up that often cocks things sideways at the crucial initial contact. Pressing things out isn't an issue. Pressing things in is where it gets frustrating.Originally I didn't like it loose either. But I got over that very quickly. I reasoned that it was better to have the frame joints flex than break. I have never needed it to be solid. Once the jack takes up the frame play, nothing moves.
I may look at a re-design of the cradle, maybe something w/ bearings. I thought about using an inverted, solid mount cylinder, but really like the convenience (and cost!) of my spare pneumatic over hydraulic one. That and, like you, I keep a couple different capacity cylinders handy, it's easy to change them out.