I think you're overthinking it. The artificer stores a spell in the SSI. How the creature activates that spell's effect is irrelevant. The spell effects all use a standard set of rules in chapter 10 of the PHB. Those are in effect until a specific change is given such as applying the artificer's INT as the ability modifier to use.
@cbwjm gave the same formula from the DMG regarding item use but that includes a helpful tips on what to do when the user doesn't have a casting stat, but it's moot given SSI covers the ability score.
All PC abilities use the same formulas to calculate DC's and attacks. It doesn't matter whether it's casting a spell, ki stunning an opponent, manuever disarming an opponent, or activating a spell from an item.
The only differences are it's a different action type to do it with the SSI and the default calculation from chapter 10 use the artificer INT as the caster ability score. The spell effects all use the same resolution they normally use because the SSI hasn't changed those.
I agree the general vs specific gets muddled up, but not in this case.