When I look at the class and its spell progression, it looks like a 1/2 caster since it gets spell slots at 1/2 the progression of full casters. However, I see most people refer to it as a 2/3 caster instead. I've even seen people call it things like a 6/10 caster, but mostly it's seen as a 2/3 caster.

What exactly causes artificer to be considered a 2/3 caster rather than a 1/2 caster? Is it just because of the multiclass spell slot progression, or is there something else I'm missing?

I think that when people refer to Artificers as a 2/3 caster, they are referring to aspects of their spellcasting that set them apart from other half casters (the Paladin and Ranger)

1. The Artificer gets cantrips as part of their Spellcasting. For a Ranger or Paladin, getting cantrips requires you to choose a specific fighting style

2. The Artificer can cast spells as ritual. Neither of the other half casters are able to do this as part of their spellcasting kit. In fact, the only spellcasters that can inherently cast spells as rituals are bards, clerics, druids, and wizards (all full casters). 

Edit: On the subject of point 1, I kind of wish Tashas had made the Blessed Warrior and Druidic Warrior fighting styles  to tack on the same type of cantrip scaling the Artificer gets. It would be unorthodox, since Fighting Styles generally dont change with level, but I feel like granting the Paladin/Ranger an extra cantrip at 10th and 14th level wouldn't be overpowered by any measure. 

Quote from ArntItheBest >>

Artificers also get spells on odd levels, which means they can cast spells at level 1. No other half caster (Paladin or Ranger) can cast spells until level 2. This means that in the multiclassing table, you get half your Artificer level rounded UP instead of down, like Paladin and Ranger.

 Agreed, this also means that is a full caster makes a single level dip into artificer they do not sacrifice any spell slots. Makes 1 Artificer / X Wizard attractive for proficiency in armor and con saves.

What Kaboom979, ArntItheBest, and Jegpeg all said.👆  

PS- 2/3 ≈ 6/10 as far as anyone is concerned in this instance. I say 6/10 because they get to cast spells of 6 out of the 10 levels of spells. Remember, cantrips are 0th-level spells, so there are 10 levels of spells, not just 9 levels of them. Paladins & Rangers get to cast 1st-5th-level spells, for 5/10 or “1/2” of the levels. Artificers can cast 0th-5th-level spells, “6/10” of the levels. Adding in Ritual casting too makes it easy to round them up to “2/3” casters for simplicity.  

Ritual casting and the SSI go a long way towards helping with spell casting.

I'd say that the Artificer's basic spellcasting framework is Half+, but they're more half caster than not.

The Alchemist is the only true 2/3rds caster of the Artificers, as their Elixirs have spell equivalent effects (so depending on what you need the random Elixirs are like a free spell slot), and at later levels they straight up get bonus castings of specific spells, which brings them up to similar "slot" numbers to a full caster but without the higher level spells. While these can be situational, as spell slot equivalents it all adds up to a lot more spellcasting potential than any other half-caster.