Hi lostfan01,
Apple Music streams are music files which are from Apple Music.
If you have purchased an album from the iTunes Store, and you're playing the files from that purchased album, that is not a stream. It does count as a purchase for the artist.
If you add the album from Apple Music, and play the Apple Music files, that is a stream. Anytime you play a song from the Apple Music Library it is a stream.
This information explains what is included in the Apple Music data of an artist:
"Plays, Listeners, Shazams, and More
We measure your music's performance in these categories:
- Plays are recorded when a user initiates song playback in Apple Music for more than 30 seconds.
- Average Daily Listeners are the average number of daily unique listeners in the time frame selected.
- Purchases are recorded when a customer buys your song or album from the iTunes Store.
- Shazams are recorded anytime your song is recognized by the Shazam app or one of the partner apps that use Shazam technology.
- Insights are a list of milestones and all-time bests for your songs and albums, highlighting important changes in the data.
- Radio Spins are recorded when a subscriber listens to your song for more than 30 seconds while tuned in to an Apple Music radio station. Apple Music radio stations are created when a user picks a song or an artist to generate a custom radio station, or from an Apple Music curated radio station.
- Video Views are recorded when an Apple Music user watches your music video for more than 30 seconds."
You can find this and more here: Understand Your Analytics - Apple Music for Artists.
A stream will count as a "Play" no matter where you're playing it from if it's an Apple Music file. It can be part of a playlist, from a search, and while playing a file offline - so long as these are Apple Music files. If they're files that you purchased, it would be part of the "Purchases" category.
There are content providers who the artists work with. This page provides more information: iTunes Connect Resources and Help.
Cheers!