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

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… [ERLW p. 58]

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

These spells count as artificer spells for you… [ERLW p. 58, 59, & 60]

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
  • Artillerist: Thunderwave, Scorching Ray, Shatter
  • Battle Smith: Heroism, Warding Bond