You use the spellcasting ability of the original spellcaster
You have quoted the relevant rules text from the Ring of Spell Storing (emphasis mine):
[...] While wearing this ring, you can cast any spell stored in it. The spell uses the slot level, spell save DC, spell attack bonus, and spellcasting ability of the original caster, but is otherwise treated as if you cast the spell. [...]
This is a clear cut case of specific vs general:
Remember this: If a specific rule contradicts a general rule, the specific rule wins.
The general rules for casting a spell, as specified in the Casting a Spell section of the PHB Spellcasting chapter, are:
When a character casts any spell, the same basic rules are followed, regardless of the character's class or the spell's effects.
[...]
Many spells specify that a target can make a saving throw to avoid some or all of a spell's effects. The spell specifies the ability that the target uses for the save and what happens on a success or failure.
The DC to resist one of your spells equals 8 + your spellcasting ability modifier + your proficiency bonus + any special modifiers.
[...]
Some spells require the caster to make an attack roll to determine whether the spell effect hits the intended target. Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.
So, for the purposes of any spell cast using this magic item, the spellcasting ability (and consequently the spellcasting ability modifier) is changed from your normal spellcasting ability to that of the original caster.
What does this mean for the features you mentioned?
Casting Counterspell from the ring
Counterspell states:
[...] If it is casting a spell of 4th level or higher, make an ability check using your spellcasting ability. [...]
But, for the purposes of Counterspell cast using a Ring of Spell Storing, the spellcasting ability is replaced from being your spellcasting ability, to that of the original caster. As a result, we use the spellcasting ability of the original caster to make this check.
Casting Dispel Magic from the ring
Dispel Magic is affected in precisely the same way, given the text of required check is the same (to a reasonable degree of accuracy):
[...] For each spell of 4th level or higher on the target, make an ability check using your spellcasting ability. [...]
Arcane Trickster's Spell Thief feature
This feature needs a bit more parsing, and as a result, I'm going to quote more of the feature's text than you have (emphasis mine):
At 17th level, you gain the ability to magically steal the knowledge of how to cast a spell from another spellcaster.
Immediately after a creature casts a spell that targets you or includes you in its area of effect, you can use your reaction to force the creature to make a saving throw with its spellcasting ability modifier. The DC equals your spell save DC. On a failed save, you negate the spell’s effect against you, and you steal the knowledge of the spell if it is at least 1st level and of a level you can cast (it doesn’t need to be a wizard spell). For the next 8 hours, you know the spell and can cast it using your spell slots. The creature can’t cast that spell until the 8 hours have passed. [...]
I'm also, for ease of reference going to requote the relevant text from the Ring of Spell Storing (with a slightly different emphasis):
[...] While wearing this ring, you can cast any spell stored in it. The spell uses the slot level, spell save DC, spell attack bonus, and spellcasting ability of the original caster, but is otherwise treated as if you cast the spell. [...]
So, in this case, the casting of the spell has completed, and the Arcane Trickster is targeting you with it's ability, not the Ring of Spell Storing. So you need to make a check with your spellcasting ability.
This also has some implications for both you and the ring:
- The spell that was cast from the ring cannot be cast by you (either using your own spell slots, or another charge of the spell in the ring). This is specifically because the ring explicitly says that the spell is treated as if you cast the spell.
- So, if it's a spell that is on your classes spell list, or is a spell you can cast using a racial trait, class feature, or other game feature, then you can't cast the spell that was stolen under any of those features.
- This restriction lasts for 8 hours
