The first set of terminology, with intros, buildups, and drops, is a starting point most useful for describing a range of dance music that is instrumental (no vocals) or instrumental-driven (has vocals, but these are an element of the track rather than the sole focus).
The second set of terminology (verse, chorus...) is a set of terms to describe the structure of more typical vocal-driven music, where the lyrics have a lot to say and the lyrical progression drives the song forward more than the instrumental arrangement.
Of course both the particular structures you mentioned are only examples. Dance music tends to have a number of different arrangements of a track, for different environments - radio play and club play by a DJ, for example. A club mix would typically be longer and give the DJ more options for mixing in and out of the track, while a radio mix has to condense the important elements into a shorter space of time. There are also plenty of 'traditional' songs that follow different structures to the example you quoted.
Also, as electronic dance has become more popular and mainstream, there are many tracks that use electronic instrumentation but follow a somewhat verse-chorus based structure - or something in between. If I recall correctly Kiesza - Hideaway and Rihanna - We Found Love are two that have traditional song-style verses, but instrumental choruses.
So if you have no vocals at all, you may find it more helpful to think in terms of "Intro-Breakdown-Buildup-Drop-etc." If your vocals follow a Verse-Chorus-Verse-Chorus pattern, that may be a more helpful way to think of things - or you could aim for a structure somewhere in between. If you are starting out, you could maybe try a few different arrangements of your track to see if you can get it to work in different ways.