TIM will honor quantity-limited requests before doing refinery management, so one strategy to sequester your stone is to tag a container like "[TIM Ore/Stone:1000000]". Because that request has a quantity, TIM will try to put up to a million kg of stone in there before doing anything else with it; you could then connect that container to an ejector if you want to dump the stone.

tried that, with the exact tag you listed, but the container was not collecting any stone as it was going to refinery instead, even tried to set priority P1 which didnt help either

I'm thinking it could be the default refinery behaviour gaining priority over TIM somehow, but dunno

also what might be important and I forgot to list it, this was happening on a dedicated server (modless afaik)

I solved it his way:
1.) Drills put everything into a storage array, all containers named [TIM]
2.) Refinarys are named [TIM Auto] and are connected to the storage ONLY through two sorters, ->Refinary blacklist stone, Refinary-> whitelist stone. No need to make the sortes activly pull anything, TIM handles it.

This works fine so far.

3.) To eject stone, put an ejector at the end of the storage (not where the mined ores comes in!) and name it [TIM Stone:P10]. It will eject perfectly. For savety proposes you can put a sorter to the ejector and whitlist Stone.
4.) To keep some stone at stock, just name one container of the storage array [TIM Stone:P5:10000]

5.) If you are using a Stone Crusher mod, just connect the crushers directly to the storage array. Let them work without TIM or name them all [TIM Stone:P1], so TIM can evenly make all crushers work. In that case you only will have gravel left, To eject this just rename the ejector to [TIM Ingot/Stone:P10] and the stockpile-container to [TIM Ingot/Stone:P5:10000] (Dont forget the to change the whitliet of the savety-sorter for the ejector)

Hope it helps.