Multiple prestige classes (or, what's an "arcane spellcasting class"?)

Say Jim-bob is a Wizard 3/Cleric 3/Fighter 1. Jim-bob then takes a level in Mystic Theurge, gaining 4th (class) level wizard and cleric spells per day. Jim-bob then takes two levels in Eldritch Knight, the second of which grants him "new spells per day as if he had also gained a level in an arcane spellcasting class he belonged to before adding the prestige class".

Can Jim-bob choose Mystic Theurge as his arcane spellcasting class for Eldritch Knight, gaining both wizard and cleric spells? It's a class that casts arcane spells, after all.

No. A Mystic Theurge doesn't cast spells as a class feature -- it adds ability to a class (in this case two classes) for the purposes of casting spells from that class's class features.

You don't have a "mystic theurge casting level", "mystic theurge spells", or a "mystic theurge spell list" -- as such it is not a caster in its own right and not a choice for the purposes of increasing spell ability from another prestige class.

The wording isn't the best, almost every reference to class in the PrC chapter should be "core class".

Arcane Spellcasting Class is meant to be a class with solid and clear arcane spell casting features (i.e. Use Magic Device, Rogue Magic Talents and such doesn't count), JJ clarified this.

Finally Mystic Theurge cast divine spells too, not sure if it can be defined as an arcane spellcasting class, or anything at all.