Do I need explicit clearance to taxi on an inactive runway?
Yes, you need an explicit instruction (not clearance) to taxi on any runway - and you got one!

The controller told you to "taxi on [runway] 10" which is about as explicit as it gets. If he did not want you to taxi onto the runway, he would have explicitly told you to hold short of runway 10. Note that it doesn't really matter if the runway is "active" or not. In the eyes of ATC, a runway is always a runway. Don't expect the controller to say something like "cleared to enter runway xx" - because it is not a clearance, but an instruction. The phraseology used will be along the lines of "enter runway xx", "taxi on runway xx", "cross runway xx" etc. - never "cleared". The only time you will hear "cleared" and "runway" in the same sentence is in a takeoff clearance or landing clearance.

That said, you did exactly the right thing. If in doubt, stop and ask. Especially when it comes to runways, if in doubt, never assume you have been instructed to enter a runway. Runway incursions are very dangerous, and the controller would much rather clarify an instruction instead of risking a collision.