Does a cantrip need to be learned like level 1-9 spells as a wizard?

Question

Upon hitting level 4, Wizards go from possessing 3 cantrips to 4. Additionally, "Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free." (from the PHB)

Does the cantrip gained at level 4 take one of these two "learned a new spell" slots?

I know cantrips are treated differently for loads of cases, and I'm not sure if this is one of them. Not a big deal either way I just would like to make sure I'm doing this correctly.

Which of the following interpretations is correct?

• Option A (possibly illegal):
  Upon hitting level four, the wizard learned Tenser's Floating Disk, Unseen Servant, and Prestidigitation.
• Option B (definitely legal, but may not be the correct move):
  Upon hitting level four, the wizard picked up Tenser's Floating Disk and Prestidigitation.

Answer 1

Cantrips do not count against the two spells you learn when you level up

The Wizard's Spellcasting feature states:

Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free [...]

However, it is important to note that this is under the section "Learning Spells of 1st Level and Higher", which tells us that this applies when learning spells of 1st and higher - and thus does not apply to cantrips, which are level 0 spells:

Every spell has a level from 0 to 9 [...] Cantrips -- simple but powerful spells that characters can cast almost by rote -- are level 0 [...]

In fact, the wizard's section on "Cantrips" states:

At 1st level, you know three cantrips of your choice from the wizard spell list. You learn additional wizard cantrips of your choice at higher levels, as shown in the Cantrips Known column of the Wizard table.

Looking to the Wizard table, we see that at level four you go from knowing three cantrips to knowing four, and so you end up learning one additional cantrip upon reaching level 4. This does not impact the two spells gained at any given Wizard level-up because cantrips are not spells of 1st level or higher.

Something else worth noting is that cantrips are not part of your spellbook at all, so they could not possibly be the two spells you add to your spellbook:

[...] Your spellbook is the repository of the wizard spells you know, except your cantrips, which are fixed in your mind.