There was "I Love Lucy." (Lucy Ricardo)
"The Lucille Ball-Desi Arnaz Show," which was basically the "I Love Lucy" show in an hour fomat each week built around a guest star. This had lots of continuity issues from the "I Love Lucy" show, especially the first show guest starring "Ann Sothern."
Then you have "The Lucy Show," (Lucy Carmichael) which was one show but was really two shows, when she was in Danfield NY, then when the show moved location to Los Angeles, CA, most of the continuity was lost and it really became a new show.
Then you had "Here's Lucy," (Lucy Carter), which was an entire new show. Lucille Ball had sold DesiLu during the last year and wanted to have a new show directly under control, which was Lucille Ball productions which was a subsidary of the company Gulf+Western which bought DesiLu.
That show not only had continuity errors but script problems because the writers seemed to forget that on this show Gale Gordon who played Harrison Cater, her employer/brother-in-law, was her relative. The writers would often screw that up by failing to realize relatives of Lucy's kids from her late husband were ALSO relatives of Harrison Carter
But that is more poor writing than continuity
The reason people get confused is basically though the location and premise was different Lucy was always based on the shame "Lucy" character created by Jess Oppenheimer, for the "I Love Lucy" show. In fact he sued and won money, saying Lucy Carmichael and Lucy Carter were in fact the same character as Lucy Ricardo, just thinly disguised. (his original contract entitled him to a percentage of the "I Love Lucy" show and characters created.)
Trivia, Lucille Ball had a superstition and thought the letter "AR" we lucky that is why her characters were ricARdo, cARmichael, cARter. Lucy herself when asked why she thought that because she was BALL, she replied, "But I didn't become successful till I became lucy ARnaz.