The spell casting rules in the PHB govern your spells which would excludes those rules as well if the spells are deemed to be originating from the SSI which is the base of the "it's not spell casting" ruling which has been verified by sage advice.
The only restrictions the SSI have to be activated is it needs to be held by a creature that can take an action,which the SSI feature provides if needed, and the spell effect(s) occur. The spell casting ability modifier is given but the rest is omitted which leaves room for multiple interpretations on how to fill in the blanks or if the blanks exist at all.
The SSI is an paradox of sort of you look at it without infering what they meant
So far as I know, there is no general rule for spell effects produced from items. A group might feel the SSI to be similar to a ring of spell storing, which users the original caster's stats. One puzzle will be why SSI references only the artificer's ability modifier? Elsewhere, where such limited language is found, it means what it says: use only the ability modifier, not proficiency. Still, most (perhaps all?) items - from scrolls to rings to wands - that store spells that are later cast or produced from them do use values that match an original caster's stats. If there is an intended general rule, then I believe that provides relevant context.
If I understand you correctly, you believe the ability modifier is the artificer's, and that PB is going to be used, but that it is going to be the SSI's current wielder's? That doesn't match any general rule at all, not even that for a spellcaster casting their own spells. I'm trying to think of a case where a spell cast or production mixes one creature's ability modifier with another's PB... can you point to any?
I'll illustrate to save time. Bob the artificer (alchemist) turns 11th level and wants to use the SSI.
Spell-Storing Item said:At 11th level, you learn how to store a spell in an object. Whenever you finish a long rest, you can touch one simple or martial weapon or one item that you can use as a spellcasting focus, and you store a spell in it, choosing a 1st- or 2nd-level spell from the artificer spell list that requires 1 action to cast (you needn’t have it prepared).
While holding the object, a creature can take an action to produce the spell’s effect from it, using your spellcasting ability modifier. If the spell requires concentration, the creature must concentrate. The spell stays in the object until it’s been used a number of times equal to twice your Intelligence modifier (minimum of twice) or until you use this feature again to store a spell in an object.
Bob looks at the spell list and decides levitate is a good option.
Levitate said:One creature or loose object of your choice that you can see within range rises vertically, up to 20 feet, and remains suspended there for the duration. The spell can levitate a target that weighs up to 500 pounds. An unwilling creature that succeeds on a Constitution saving throw is unaffected.
The effect for levitate includes a CON save so Bob needs the rule for the saving throw that's included in that part of the spell effect. Bob looks in chapter 10 of his PHB and finds the rule.
PHB Chapter 10 - Saving Throws said:Many spells specify that the target make a saving throw to avoid some or all of the 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 any one of your spells equals 8 + plus your spell casting ability modifier + your proficiency bonus + any special modifiers.
Bob gives the SSI to Marley the fighter. Marley uses the SSI to levitate the evil assassin Dargoon. Marley looks up how this works in his PHB and sees what is quoted above. The spell effect for levitate states the target is required to make a CON save. The rules for spell casting in chapter 10 of the PHB state the formula is 8 + 4 (Marley's proficiency bonus as an 11th level fighter) + 5 (Bob's INT bonus as per the SSI modification to the general formula found in chapter 10 to which levitate refers) for a 17 DC save requirement.
Marley also notices the spell attack rules and formula are right underneath the saving throw rules in the same section. ;-)
It's RAW because all of the spell effects that the SSI would use tell the player how to apply the INT modifier in the spell description, and those spell descriptions use the general rules for DC's and attacks given.
If the SSI was cure wound instead Marley would heal someone d8+5 because the spell dictates how the ability modifier is applied. The SSI only dictates which ability modifier to use; nothing more.
Bob casting cure wounds (as an alchemist) would heal d8+10 using alchemist supplies while a typical bard would use the same slot for d8+5 getting back to our comparisons.