The rim is off, the lid is secure, but the button can move up and down, is it safe?
The button on a metal lid- whether purchased or home canned- is an indicator of a vacuum in the bottle.

During canning, a vacuum is formed that sucks the lid on tight and pulls in the button so it can't be depressed.

If the button can be "popped" then there is no vacuum and the bottle is not sealed. If the lid doesn't move that just means it's stuck not that it is sealed.

This means that the safety of the food is compromised. If you just canned the bottle then you can either try to can it again with a new lid or put it in the fridge to consume immediately. If it has been in your pantry then consider it spoiled and throw it out.

Incidentally- having a button not pop does not guarantee that the bottle is sealed. Rarely, the button will be sucked in and the seal will fail but the button does not pop back out. If the lid is stuck in place you would discover this when opening the bottle as there would be no resistance from a vacuum. After bottling, I gently check the seals on my lids even if the button is in.