You're correct- nowhere in the Core Rulebook does it make any distinction between different kinds of magic, and nowhere in the Core Rulebook does it say that you have to use a specific class' spell slots to cast those spells. In fact, I could only find this on determining what your class controls for spellcasting:
Your class’s Spells section describes which class’s spell list you
can choose from, how to determine the number of spells you
know, and at which levels you can learn new spells.
You'll note it does not say that the Spells section for your class determines the total number of spells per day you can cast.
There is also some evidence (not a lot) to support the theory that casting can be mixed. From the section on Multiclassing:
Instead of gaining the next level of
his existing class, he can add a level of a new class, adding all
the 1st-level class features of that class to his existing class
features. This is referred to as “multiclassing.”
is an exception—when
determining caster level,
a character adds together
his levels from different
spellcasting classes (such as
mystic and technomancer).
So, while there is a lot of grey area in which spell slots you can use to cast which spells, your level 1 mystic / level 1 technomancer is definitely a level 2 caster.
That's RAW. If we're going to open up the rules to interpretation (and we currently have to), I'd argue in your favor. Multiclassing in the fashion you're discussing would cause you to have a large number of lower level spells but would force you to wait twice as long before earning the higher level spells, which seems a fair trade. There's also some wordage buried in the various descriptions which implies that you normally combine everything but class features when multiclassing... and spells per day are not considered "class features" in and of themselves (page 59, though that may just be poor wording choice).
Of course, there's always the safe option- a number of spells are on both class spell lists, such as detect radiation and life bubble. If you're still in doubt, stick to casting those.