Are both diagrams correct? Can I say that the muon neutrino can emit a $W^+$ boson that the down quark absorbs?
They are the same diagram as far as the prescription for writing the integral that has to be evaluated for the interaction. The arrow on charged particles changes the particle to an antiparticle, so a W+ with the down arrow is a W-, by the definition of the rules of the feynman diagrams.
I would read it as "a virtual W- emitted by the down quark changes it into an up quark and changes the muon neutrino into a muon".
Or one can read it as "a virtual W- is emmited by the muon netrino turning it into a mu- , and the down quark absorbs it and turns into an up quark.
Charges are conserved in either interpretation, and as the exchanges can only be virtual there is no problem of "decay, either way, as the masses do not allow on mass shell W's.