Plex server version: 1.31.1.6716
Player version: 9.16.1.38616 (Android TV on nVidia Shield)
I have several standard playlists. If I want to start the playlist in the middle somewhere, I scroll down to the track where I want to start and click on it. The track starts playing, but when it finishes, it goes to the next track on the album the track comes from, rather than the next track in the playlist.
First part of playlist:

Play queue after starting 3rd track in Playlist (note that the play queue is not the tracks from the playlist):

Plex Web does not behave this way. It (correctly, IMO) plays the track I selected, then queues up the rest of the playlist.