The cathode is the electrode at which the net electrode chemical reaction is reduction: the reacting chemical species gains Electrons. This definition is independent of terminal sign. During Daniell cell discharge, cations accept Electrons arriving at the copper cathode and become neutral copper. The cathode is then the positive terminal. In an externally driven electrolytic cell, the source's negative terminal supplies Electrons to the negative cathode. See Pingarrón et al., §3.4: cathode and Daniell cell.
An ion carrying electric current need not undergo reduction at the cathode. For example, in the ideal Daniell cell charge-balance picture, potassium cations arriving from a salt bridge remain dissolved while cations are consumed. Follow the half-reaction to identify the reacting species; follow electric charge conservation to account for the other moving ions.

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