Anodes and cathodes

Clarification, please.

I have always used the words anode and cathode in the context of cathode-ray tubes and electrolysis where the cathode is the source of electrons directed towards the positively charged anode.

I have just been reading the Outlook section of a recent copy of Nature and it describes the external terminals of a battery as negative for anode and positive for cathode, which hurt my head a bit. But I think the key is in the following description: inside a Li+ battery, Li+ ions move internally from the anode towards the cathode. Moving Li+ from the anode must leave it -ve, so from the outside user's perspective the anode is -ve. From the perspective of the inside the battery, the usage of anode and cathode is the same as in a CRT or an electrolytic system.

Do I have that right?

I was taught about batteries at school, but I don't think the terms anode and cathode were used in that context, hence it coming as a surprise now and, interestingly, BSM Jr at 16yrs has similarly not heard the terms used of battery structures.

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