From the personal license, you can only activate once from a site. So in your case, that should your live server. Don't activate them on your test and staging server. Activation doesn't block or remove any feature, only you get the error notice in admin panel and it's necessary for getting update.
The plugin seemed to stop working when I moved the site to a different server – but you're saying that I should not attempt to re-activate and it will work as expected? Will I be able to activate once I move the site to the final live server?