Woah, demons are like people. They are not defined either by how they think, or how they look. They just are. Just as we are. However since demons are creatures created in a fantasy sort of world, we have so many kinds, from giant spiders, to lizards, beings made completely of lighting, animated buildings, and so much more.
Do all of those transform into some sort of super-powered mode? No. But some actually do. Is it a Devil Trigger for them? NO.
Why not? Think of what Devil Trigger means and where it came from. It was created because the main hero, Dante, who isn't a complete devil, but only a half-breed can assume a demonic form for a few minutes(?). He triggers the demonic blood that runs through his human body, to transform himself into a demonic aspect of his normal self. In fact, since hhe isn't a real demon, his appearance and powers tend to vary according to the devil arms/amulets/stuff he possesses. He is something extraordinary. Same with Vergil. And same with Lucia. Nero is something close to that. He supposedly has demonic blood in him, but probably less than Dante. So, what did he do? He used Yamato, as a catalyst to create a shadow/demonic aspect of himself. He can't transform, as he's mostly human. In fact, without Yamato, he probablly wouldn't even know he was capable of doing this kind of feat.
Trish is 100% demon. She teleports/shoots lighting/jumps higher than Dante/and does who knows what else. She doesn't need to TRIGGER her power. It's always active at full. Only weird cases like Dante have the power to do that. And the same with Sparda. In the end his true form is the demonic one. Even his shadow won't change to that of a human, showing us that his human form is some sort of illusion/or imperfect transformation. Nothing like Dante's version.
As for how both Trish/Sparda are in the game? They aren't. They're just COSTUMES. They have a "DT" because the gameplay uses it, but that's that. In the end, all they have is some sort of power-up so they could be used like the normal playable characters in the game and enter a mode where they can replenish their vitality/fly etc.