A theme ain't what you do, it's the way that you do it.
Tain't What You Do - Jimmy Lunceford - YouTube
Your class is what you do.
Your theme is how you do it.
Your background is why you do it.
So..
Class: Power Ranger (I'm a power ranger)
Theme: Morphing Grid (How I do it... I morph)
Background: Why I do it is tricky... because I'm a good guy? What skill bonuses do I get for being a "good guy"?
It seems a little silly to me to break it up into these categories. Especially when the bloat for ALL THREE forms are going to go skyhigh if 5e lasts more than a few years.
It'll be..
Class: Thayan Summoner
Theme: Uberlich
Background: Wonderdog
Without any idea what those things even mean anymore. At least with 'Thayan Summoner' it should be relatively easy to look up.
[MENTION=11586]RigaMortus2[/MENTION] I think the problem is that you didn't propose an alternative. You only asked what the three roles mean.
If you had suggested something like..
Class: Basic abilities and what you do. Ie. Fighter who fights, using X, Y, Z skills to do it. XYZ will change from character to character, and even more so while they evolve and level. It would likely have some hierarchy of how or why but it should be fairly free form.
So keep it more or less as is, but add in other things too. Like a lot of what is already tied into background, and then split the other aspects from theme.
Theme: Your combat role with advanced abilities relating to it. Ie. Tank/Defender/Guardian being someone who makes sure people don't get hit. The Striker/Slayer/Berserker theme being someone who goes up and swings at the enemy until they are dead. Etc. These themes will be limited to ones created and will give a set advantage. Possibly akin to variants, kits, archetypes of previous editions. They may complete or replace the XYZ of above.
To an extent this keeps it "how" you do something without having the multiform that 5e already seems to have with lurker, warrior, tinker, taylor, thief forms.
Background: 'You' specific abilities. Things you have picked up, which shouldn't be just limited to a specific background but instead a little more freeform from the get go. You are a knight, but you don't have diplomacy you have intimidate. You picked knight so you get a combat feat, but the feat isn't set. Backgrounds should be fluid with a lot of specialty allowing two knight-fighter-slayers to be utterly different based on background alone. This would be like having a dragonmark from a house, or other social related skills and feats.
This would override the "why" you do it and more about "who" you are. I would preferably not even want to see "knight" titles at all. I would like you to be able to call yourself knight with knight-y skills being tied into the fighter class as variants. And leave background open so people can pick things that directly relate to them as a character instead of predefined packages of traits.