Lorehold Student - are the spells considered class spells.

[8wGremlin]

With Strixhaven Initiate (lorehold) you gain access to 2 cantrips and a 1st-level cleric or wizard spell.

Are these spells considered class spells for you now, as you only have the one class, and the feat allows you to cast them using your own slots.

I ask as this would be a way for the Artificer (artillerist) to get Magic Missile on their spell list.

[Dualight]

I do not have the book and thus not the exact text, so I am limited to the information you provide, but if the feat says you take the spell from a class list, it counts as a spell from that class, so, as I understand it, magic missile would be a wizard spell, not an artificer spell.
Again, it depends on the exact text, but from what I gather the Strixhaven Initate feats are variants on Magic Initiate, and the above is how Magic Initiate works, as far as I can tell.

[tokek]

It seems not. The background adds spells to your spellcasting class (or all such classes if you multi-class) but the feat does not.

[Skyrender]

Not from the expanded spell lists from any of the backgrounds, but most versions of the Strixhaven initiate feat (all except Slyth... I mean, Silverquill) have the option of picking a wizard or sorcerer spell. You could certainly choose magic missile with this option, although it's unnecessary if the artificer is going to take the armorer subclass at level 3, because that's on the armorer's expanded spell list.

[Khrysaes]

Not from the expanded spell lists from any of the backgrounds, but most versions of the Strixhaven initiate feat (all except Slyth... I mean, Silverquill) have the option of picking a wizard or sorcerer spell. You could certainly choose magic missile with this option, although it's unnecessary if the artificer is going to take the armorer subclass at level 3, because that's on the armorer's expanded spell list.

Then if the spell is on both lists, do you pick which spell it classifies as? Certain abilities or items specify a class spell, such as evocation wizard 10? That adds int to a roll for a wizard spell.

[Skyrender]

Armorer absolutely gets it as a class spell. It is literally a class feature and a class spell, and is always prepared.

The Strixhaven initiate feat allows you to cast it once without a slot, and subsequently with whatever slots you have, including the ability to pick whether you use intelligence, wisdom, or charisma as your spellcasting stat. This is different from magic initiate, which doesn't allow casting with slots you have from your class, unless you're a member of the class you chose the spell from, and requires you to use the assigned class's spellcasting stat. You could ask your DM for a ruling, but the intent feels pretty clear, given the way they structured these backgrounds: they add spells to what you can choose from, regardless of what your spellcasting (or pact magic) class happens to be. The feat is, in essence, an extension of this, allowing you to choose a single spell from the stereotypical classes for your house and apply that choice effectively.

I personally like to think the different houses went to all the trouble of researching alternate versions of each spell for each class, and once you pledge with them, you get access to those spells because they teach it in their equivalent of Spellcasting 101. For artificers, maybe it takes the form of secret blueprints for how to make a steampunk invention that does specialized magic that is so counterintuitive that nobody would intentionally discover it.