Can the artificer's spell-storing item store their specialist spells?

Question

The artificer's "Spell-Storing Item" feature says:

... you store a spell in it, choosing a 1st- or 2nd-level spell from the artificer spell list that requires one action to cast... [TCOE pg. 13]

Each of the artificer specialists has a supplemental set of spells which says:

These spells count as artificer spells for you... [TCOE pg. 14, 15, 17, & 19]

If your specialist spells "count as artificer spells for you," does that mean they're now included in your artificer spell list, so may be eligible to be used with the spell-storing item? Or because the text refers to the artificer spell list, is only the base list eligible?

The following are the spells that would be allowed in the first interpretation but not in the second:

• Alchemist: Ray of Sickness, Flaming Sphere, Melf's Acid Arrow
• Armorer: Magic Missile, Thunderwave, Mirror Image, Shatter
• Artillerist: Thunderwave, Scorching Ray, Shatter
• Battle Smith: Heroism, Warding Bond

Answer 1

Yes, specialist spells are Artificer class spells for that character.

The whole sentence is:

These spells count as artificer spells for you, but they don’t count against the number of artificer spells you prepare.

There are no additional caveats on the sentence that would exclude them from being artificer spells in a particular context. It's not "counts as artificer spells except for..." If they are artificer spells, they're on the class spell list. If you were to write down a list of all the spells the character has access to, the character's artificer spell list, you'd include them, wouldn't you?

More evidence comes from the Bard section of the Sage Advice Compendium:

Which spell scrolls can bards understand — spells from the bard list only, or spells from the bard list plus spells from Magical Secrets?

A bard can use any spell scroll that has a bard spell on it — including spells gained from the Magical Secrets feature, which are treated as bard spells for that character.

Spell Scrolls says:

A spell scroll bears the words of a single spell, written in a mystical cipher. If the spell is on your class’s spell list, you can read the scroll and cast its spell without providing any material components. Otherwise, the scroll is unintelligible.

In short, a bard spell is a bard spell, no matter how it became one. There is no difference between "bard spells" and "spells on the bard list" for a character that gains them via Magical Secrets. Likewise, there is no difference between "artificer spells" and "spells on the artificer list" for a character that gains them via a subclass.

Why is Warlock worded differently?

Cleric Domain Spells, Druid Circle Spells, Paladin Oath Spells, and so on are automatically known-and/or-prepared spells as well. The Warlock has different wording because the spells become available for selection, but are not automatically known. A warlock must pick them off the list as part of the normal level-up process - they're added to the list of choices.

...lets you choose from an expanded list of spells when you learn a warlock spell. The following spells are added to the warlock spell list for you.

Answer 2

No, the artificer cannot store their specialist spells.

You've found the crux of why they can't already. No where in the specialist spells feature does it say they are added to the artificer spell list, so they aren't eligible for the Spell-Storing Item feature. Compare this to something like the extended spell list of a warlock's patrons, where the feature says:

The [patron] lets you choose from an expanded list of spells when you learn a warlock spell. The following spells are added to the warlock spell list for you. [PHB p.108, 109, & 110]

There's no subclass that uses the same wording of ERLW, but that's because for most subclasses that have extra spells (Cleric domains, Circle of Land Druids, Paladin Oaths) have a default part in the main class saying, "If you gain "a subclass" spell that doesn't appear on the [class] spell list, the spell is nonetheless a [class] spell for you."