Your instinct is correct: the artificer is more like a cleric than a wizard when it comes to spells known. They prepare spells daily from the complete list available (which includes subclass spells).
The mechanism for changing the spells prepared is at the bottom of page 11 of TCOE:
You can change your list of prepared spells when you finish a long rest. Preparing a new list of artificer spells requires time spent tinkering with your spellcasting focus: at least one minute per spell level for each spell on your list.
The flavor of this is that the artificer is modifying their chosen objects (e.g. substances, tools, or devices) to cast the new spells. For example, maybe a mechanical spider used to spit venom for a poison attack, but now it will weave webs to close wounds for a healing spell.
Note that the similarity between artificer and cleric casting extends to ritual spells. Both class descriptions say:
You can cast [a cleric / an artificer] spell as a ritual if that spell has the ritual tag and you have the spell prepared.
The wizard, though, is different:
You can cast a wizard spell as a ritual if that spell has the ritual tag and you have the spell in your spellbook. You don't need to have the spell prepared.
"Learning" an Artificer Spell
Chapter 11 of the PHB introduced the 12 original classes' spell lists by saying:
The chapter begins with the spell lists of the spellcasting classes.
When TCOE expanded other classes' spell lists, they said something like this:
The spells in the following list expand the cleric spell list in the Player's Handbook.
Unfortunately, TCOE's authors didn't follow this bare bones approach when introducing the artificer's spell list. Instead, they used the more friendly wording you quoted, which introduced the word "learn." As you've highlighted, this word is loaded based on its use with wizards' spellcasting.
Clearly, however, the artificer doesn't gain spells the same way a wizard does. For wizards:
Learning Spells of 1st Level and Higher - Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures, you might find other spells that you can add to your spellbook (see the “Your Spellbook” sidebar).
Compare this with the sorcerer, whose player chooses a subset of spells at each level:
The Spells Known column of the Sorcerer table shows when you learn more sorcerer spells of your choice. Each of these spells must be of a level for which you have spell slots. For instance, when you reach 3rd level in this class, you can learn one new spell of 1st or 2nd level.
Importantly, with the sorcerer, there's no mention of preparing spells.
The Bottom Line: For clarity, TCOE's authors should have said "when you prepare an artificer spell" instead of "when you learn an artificer spell."
Minor Caveat: Cantrips
Cantrips are the one place where "learning" spells has relevance -- though, again, just like the cleric:
At higher levels you learn additional artificer cantrips of your choice, as shown in the Cantrips Known column of the Artificer table. [TCOE, pg. 11]
You learn additional cleric cantrips of your choice at higher levels, as shown in the Cantrips Known column of the Cleric table. [PHB, pg. 58]