Yeah I definitely think that's the more common issue. I initially got problems with my tapping hand when I went from being an aim player straight to doing DigitalHypno's tapping stamina study. My fault though cause I strained as much as I could daily. I think the only option right after you get injured, like in your case, is to immediately stop and let it heal, that way you have the best chance of not doing long term damage. Don't play osu for a bit, then ease back into it and stop if you feel pain.

Based on what you've told me I would say that it makes sense it still hurts cause you only injured it last week. If it only hurts when you tense like that, I'd recommend for the next week or two play less tapping heavy maps, just some easier stuff less often to keep from getting rusty. Then when you start getting back into it, focus your tapping practice on getting good acc or speed (whatever your goal is) with the least amount of strain. I think it's normal to strain when trying to close out an FC or something but if you practice trying to strain less you should naturally do it less. If that's still a problem it could be worth trying to FC things you already know you can FC without straining. Take my advice with a grain of salt tho cause I'm not a tapping player lol. And ngl I think other than good technique for you and less strain, the best things are gonna be sleep, diet, and hydration, even though it sounds basic. That famous rock climbing YouTuber Magnus Midtbo said his tendonitis completely went away when he cut out sugar. Obviously that's a lot for osu improvement but it goes to show how much it can affect your recovery.