AbdulAlhazred
Legend
Something else of note with the whole, which class cast the fireball, is that due to the way spell preparation works, you may never see a wizard read from a spellbook. For all we know he leaves the spellbook at home when he travels after prepping his list of spells. He might still carry a ritual book which records of the ritual spells he has, but then literally any class can have a ritual book with the right feat. I find that really speaking, the lines are blurred. We have a total of 7* separate classes/subclasses that can cast fireball and it would be impossible to judge how they gained the ability to cast it. They could all look the same since even a wizard can wear platemail and still cast spells if he has spent the feats or is multi-classed (both feats and multiclassing blurs the lines between classes even more).
In my current game, I'm letting my players create the fiction of their class. If that means that they follow what's suggested in the book then I'm cool with that. Really speaking though, not everything needs to be laid bare about how it looks in the game world, I don't need to know if people can tell if that guy in robes who threw a fireball was a wizard, a sorcerer, or an eldritch knight. In the end, all that matters is that the enemy is nice and crispy.
*bard, eldritch knight, wizard, warlock, sorcerer, arcane trickster, light cleric. I may have missed some.
Heh, yeah, in our 5e campaign my wizard is a Mtn Dwarf Transmuter. He's clad in Chain armor, and carries around a battleaxe, which he's credibly good at wielding too (more for show than anything, but appearances matter to him). Not that anyone would mistake him for a non-caster, but they might easily call him an 'Eldritch Knight' or whatever. A level or two of MCing or even one or two feats would probably make that axe a better option for toe-to-toe than his Fire Bolt.