Basically just from experience
For me, if something comes out and goes back in it's mostly accompanied by the click/crunch/thump (usually with achy pain afterwards), whereas if something goes twang it's more of a sharp nerve-type pain that makes you gasp and swear - or at least it does for me!
Treatment for both is the same - stop what you're doing for a moment, give it a rub, consider changing activities or having a tea break. Washing my hands in hot-as-I-can-stand water helps sometimes. See if you can do things differently to avoid making it go ping - you're right that adding a splint and carrying on will just shift the problem elsewhere
My hand therapy came from a referral by the UCHL clinic - but you should be able to ask without anyone exploding. Sadly when I moved from the main hand therapy people into the dressing clinic after my first hand surgery they had to discharge me, then when my hand was better and the dressing clinic discharged me I was left with neither! Make a fuss if that happens