Played this a bit more and discovered a few things. The game has what it considers to be 3 different style of dungeons:
- Adventure Dungeon
- Traversal Dungeon
- Arena Dungeon
The first dungeon experience I described is considered an adventure dungeon that has a mix of fighting and puzzle solving with some back-tracking to get additional cores to progress.
A few dungeons later I came across a Traversal Dungeon. There was no fighting involved. The experience was all about platforming to get from the beginning to the end. There are 3 secondary objects for Traversal Dungeons:
- Finish it under a time of X:XX
- Collect the Yellow Key
- Activate all 8 switches
For each secondary object you complete you get a little bonus at the end of the dungeon. If you get all 3 secondaries in the same run you get a 4th bonus at the end. In the two I played it was either a mini-stash box full of items or blueprints to craft new sidekick body parts (Activate all 8 switches), or a secondary Prismatic Core (Yellow Key Collection). The activation of the switches was shooting floating bots with the appropriate weapon rifle color.
I have not experienced an Arena Dungeon yet.
EDIT: Here is a video I found which gives a brief look at some various bits of the game.
At 0:44 the world map (as you know it) is shown
At 1:07 the beginning of a Traversal Dungeon is shown as well as the 3 secondary objectives.
At 1:10 and 1:13 the activation of two Switches are shown by shooting them.
At 1:15 the finding of the Yellow Key begins.
At 1:23 the end of the dungeon and awards are presented, the first room is for completion time, the second for yellow key, and third for all 8 switch activations, with 4th award being for hitting all 3.