The spell scrolls can be renamed easier because they all do function the same way.
Can a fighter with no spellcasting classes cast a spell from a scroll? The DMG (p139) says that "Any creature that can understand a written language can read the arcane script on a scroll and attempt to activate it." The scroll mishap table (DMG, p140) says that "A creature who tries and fails to cast a spell from a spell scroll must make a DC 10 intelligence saving throw. If the saving throw fails, roll on the spell mishap table." Would I be correct in assuming that if I make the DC 10 save (as DM I would add the spell level), that I successfully cast the spell?
Assuming the scroll is not a scroll of protection, which I'm sure it isn't since your question was specifically about casting spells, the short answer is no, the fighter would not be able to use the scroll. He or she may attempt to use the scroll, as it says in the text you quoted, and would certainly fail because of the requirement that the spell be on the character's class's spell list. If you use the scroll mishap variant, which is optional, the fighter would then make the intelligence saving throw you mentioned to see if any of the described ill-effects are suffered.
The general procedure for using a scroll is this: if it's a scroll of protection, you're good. Anyone can use it. If it's a spell scroll, a character may only use the scroll if the spell it contains appears on his or her class's spell list. If not the scroll is unusable, and any attempt to read it will result in failure. If the spell is of the correct type for the caster and is of a level for which the caster has spell slots, then the spell may be cast. But if the spell is of a higher level, the caster must make an intelligence (arcana) check with a DC of 10 + the spell's level to see if the casting is successful. Only if the caster fails this check, and if you are using the rules variant, would the caster make an intelligence save to see if a mishap occurs. In any event the scroll is consumed.
It seems that if I want to cast a spell from a scroll... there's a feat for that.