TL;DR The cathode in an electrolytic process is considered to be
negative, so there is actually no contradiction. The cathode is a positive electrode in a galvanic cell.
There are different notations for the sign (±) of the cathode used in the literature, which are determined, in particular, by the nature of the process. A very broad definition of a cathode is that it is the electrode of some device connected to the negative pole of the current source.
For electrolysis it is commonly believed that the “−” cathode is the electrode on which the reduction process takes place, and the “+” anode is the one where the oxidation process takes place. When the cell works (for example, during copper refining), an external current source provides an excess of electrons (negative charge) at one of the electrodes — the cathode, where metal is reduced. On the other electrode, there is a lack of electrons and oxidation of metal takes place — this is the anode.
At the same time, during the operation of a galvanic cell (for example, copper–zinc), an excess of electrons (and a negative charge) on one of the electrodes is provided not by an external current source, but by the actual oxidation reaction of the metal (zinc dissolution). Following the definition given, this electrode is an anode. Electrons passing through the external circuit participate in the reduction of copper, so the cathode will be a positive electrode in this case.