As for which aircraft was designed as a trainer, it's pretty simple; the PA38 and the Warrior are both designed as such, they just go about it differently. And for some reason, one was only produced for 5 years, the other has been built for decades; some hint perhaps as to which is the superior trainer in most people's opinion...
Which is another reason to favor the PA28; if you're lucky you could get a Warrior III, or at least a recent Warrior II in good condition; not some old beat up wreck of a 152 or PA38 that students have been puking in for 25+ years...
On a different note, are there actually aircraft used as trainers that
are unsuitable as such? What always ends up being discussed is the relative merits of the 152/172/PA28/PA38/DA40, but so what? They are all excellent trainers, so makes no difference whatsoever...