The anode is the electrode at which the net electrode chemical reaction is oxidation: the reacting chemical species loses Electrons. This definition is independent of terminal sign. See IUPAC Gold Book: anode.
During Daniell cell discharge, a zinc chemical atom becomes a cation, leaving two Electrons in the electrode; those Electrons leave through the external electrical circuit. The zinc anode is the negative terminal. In an externally driven electrolytic cell, Electrons instead leave the anode through the lead to the source's positive terminal. Conventional current in that lead points toward the anode, opposite to Electron motion. The anode can therefore be positive while still being the site of oxidation. See Daniell cell for the balanced half-reaction and electrode for the operating-mode sign conventions.
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