Yep, the Low Cost DMM will be fine if you just want it for basic guitar work. If you're testing lots of components regularly then it's nice to have one that can also test capacitance etc.
If your tests showed that the coil pairs are actually green/white, red/black then when you re-wire the mini-toggle you'll need to swap the red and white wires on the switch from the diagram. That is, your white wire will bridge the two diagonal lugs and the red wire will go the single lug instead. That explains why you were only getting sound in the 'series' position and nothing in the other two at the start of all this.
EDIT: Before you wire up your new switch test it and compare to that type 1/2 diagram. If it is type 2 then wire it as I described above. If it's type 1 then we'll need to change the switch connections slightly.
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