There are numerous differences between the Ritual Caster feat and the Ritual Casting feature that various classes get
The Ritual Caster feat states, in full:
You have learned a number of spells that you can cast as rituals. These spells are written in a ritual book, which you must have in hand while casting one of them.
When you choose this feat, you acquire a ritual book holding two 1st-level spells of your choice. Choose one of the following classes: bard, cleric, druid, sorcerer, warlock, or wizard. You must choose your spells from that class's spell list, and the spells you choose must have the ritual tag. The class you choose also determines your spellcasting ability for these spells: Charisma for bard, sorcerer, or warlock; Wisdom for cleric or druid; or Intelligence for wizard.
If you come across a spell in written form, such as a magical spell scroll or a wizard's spellbook, you might be able to add it to your ritual book. The spell must be on the spell list for the class you chose, the spell's level can be no higher than half your level (rounded up), and it must have the ritual tag. The process of copying the spell into your ritual book takes 2 hours per level of the spell, and costs 50 gp per level. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it.
The Bard, Druid, and Wizard Ritual Casting features state:
You can cast any bard spell you know as a ritual if that spell has the ritual tag.
You can cast a druid spell as a ritual if that spell has the ritual tag and you have the spell prepared.
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.
Lets break apart the Ritual Caster feat for some information then:
- When you take the Ritual Caster feat you are automatically granted a ritual book, similar to somebody multiclassing into wizard automatically getting a spellbook. This is because the feat simply states "When you choose this feat, you acquire a ritual book".
- Ritual Caster only lets you cast the spells as rituals; you cannot cast them using your own spell slots.
- Ritual Caster explicitly requires that your ritual book be in your hand when casting the spells.
- You acquire new rituals by finding them in written form and transcribing/translating them into your ritual book.
Compare this with the Ritual Casting feature:
- There is no ritual book (unless you're a wizard; in which case you have an entire spellbook); if you needed a book, the rules would say so. A bard can just ritual cast things they know and a druid can just ritual cast things they have prepared, no books required.
- You actually can cast ritual spells using spell slots (wizard can do this only if they also prepare the spell in question).
- You aren't required to hold some book when casting rituals using the class feature.
- You acquire new rituals by actually preparing/learning new spells. Wizards are an exception as they have a spellbook and can, similar to Ritual Caster, translate/transcribe spells into their spellbook.
One place that these features are the same is that you are restricted to a class. When somebody with Ritual Caster is adding a new spell into their ritual book the new spell must be on the spell list of the class they chose when taking the feat. Similarly, a spellcaster can only cast as a ritual the spells that are associated with the class giving them the Ritual Casting feature.
Looking at your example then, when an artificer takes the Ritual Caster feat they must choose a class from this list: bard, cleric, druid, sorcerer, warlock, or wizard. You are then automatically granted a book with two 1st level ritual spells from the class that you chose. In this case, because you are taking both alarm and detect magic there is only one possible class you could choose: wizard. Thus, you can cast both of these spells as rituals while holding the book in hand and can add other wizard spells that you find in written form into the ritual book.
If you wanted to learn a spell another artificer knows they would have to first scribe the spell into a spell scroll and then you would have to decipher it and add it into your ritual book. Of note, this would have to be a wizard spell as those are the only kinds of spells you can put in your ritual book. Thus, with another artificer (or just yourself) you could add identify, magic mouth, skywrite, and water breathing into your ritual book, assuming your GM allows you to scribe spells in to a spell scroll at all.
(There's some weirdness with whether a scribed artificer spell counts as a wizard spell but that is effectively covered in this question where the answer is "Yes").
