anode and cathode in lithium ion battery?
This is indeed correct, since, as you said, the cathode is per definition the site for reduction and the anode for oxidation, both names have to be "switched" when viewing electrolysis (charging of the Li-ion battery) and galvanic cells (discharging of the Li-ion battery). What does not change is the location of the + and - pole.

A good way to remember the site of oxidation and reduction is the word pair an-ox /cat-red.