Any confusion you are having may be down to conventional vs electron flow.
In conventional flow (how people almost always think of electricity working from the "+" to the "-" of a DC circuit), the Cathode is where electrons leave [a thing] and the Anode is where electrons enter [a thing], so in a battery supplying current we consider the positive terminal the Cathode, and the negative terminal the Anode, in diodes we consider the (forward) "positive" side the Anode and the "negative" side the Cathode.
So a battery connected with a (forward) diode, we consider the electricity leaving out of the battery Cathode, entering into the diode Anode, leaving out of the diode Cathode and entering into the battery Anode.
In electron flow (how it actually physically works), the electron flow is opposite, electrons enter at the Cathode and exit at the Anode.
It is easier, unless you really really need to, to just go with it and use conventional flow, forget about electron flow, it really doesn't matter that we think of it backwards.
It is a little bit unusual to refer to the Cathode and Anode of a battery (or supply), probably because of this confusion especially since most people would most commonly use Cathode and Anode in reference to (forward) diodes.