What is classified as a "spell"?

Question

I'm curious to understand the specifics behind cards like Clarion Spirit where it states:

Whenever you cast your second spell each turn, create a 1/1 white Spirit creature token with flying.

Does playing that card count as the first spell? I know playing non-lands from the hand are spells, but what about things like Activated Abilities?

For example, what is the outcome of playing the Clarion Spirit, then upgrading my Paladin Class from Level 1 to Level 2; is that enough to trigger the creation of a Spirit Token?

This seems to indicate that a spell is something that uses the stack:

601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect.

And activated abilities use the stack:

602.2. To activate an ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect

Which is then backed up by:

602.2b The remainder of the process for activating an ability is identical to the process for casting a spell listed in rules 601.2b–i.

But at the same time they're in their own section of the rules.

So what is a spell? Is an activated ability a spell?

Answer 1

No, an ability is not a spell or vice versa. The fact that both use the stack is incidental.

The defining feature of a spell is not that it's on the stack, but that it's a card on the stack.

112.1. A spell is a card on the stack. As the first step of being cast (see rule 601, “Casting Spells”), the card becomes a spell and is moved to the top of the stack from the zone it was in, which is usually its owner’s hand. (See rule 405, “Stack.”) [..]

Some effects let you copy a card and cast it. Those also result in a spell even though it has no card associated with the spell itself. However, these effects are rare and have specific wording that you can't confuse with a simple activated ability. Most of these cards let you copy and cast cards.

112.1b Some effects allow a player to cast a copy of a card; if the player does, that copy is a spell as well. See rule 707.12.

An ability (activated or otherwise) is also on the stack, but it's never a card, only an object without a card representing it. You also never cast an activated ability: there is always another object involved which has the activatable ability on it.

113.1c An ability can be an activated or triggered ability on the stack. This kind of ability is an object. (See section 6, “Spells, Abilities, and Effects.”)

Answer 2

With Magic, you have to be very careful about your terminology. Not only what Hackworth says in his (very correct) answer, but also what "play a" means. For instance, what you actually said in your second paragraph:

Does playing that card count as the first spell?

is true (okay, if being completely correct, you "cast" spells, you "play" lands, so "Does casting that card count as the first spell?"). If you cast Clarion Spirit as your first spell of a turn (and it resolves), and then cast another spell, Clarion Spirit's ability will trigger.

Note (as the Gatherer ruling states) if you cast a spell, and then cast Clarion Spirit (and it resolves), the trigger will not happen, because the creature was not on the battlefield at cast time, when the trigger would happen.

But as Hackworth says, triggered abilities are not spells. Activating activated abilities is not casting a spell.