My experience has been that immigrants who came to the US as young children could be rather confused about where they were born or where their families last lived before immigrating or where their families originated. The best example I have is of a distant great uncle Alex Locke of Chicago, who told everyone he was born in Estonia or Germany, or even the German city of Reval (now Tallinn, Estonia). Jewishgen records tell otherwise - he was born Elia Lak in Telsiai, Lithuania (then in Russia) in 1876. By the early 1880s, his family had moved to Estonia (then in Russia), with their days in Lithuania somehow forgotten. Ten years after that, they moved to Chicago.
I see that Lodz is only about 75 miles from Warsaw. Perhaps the family did spend some time there prior to immigrating. Whatever the case may be, the records you have found for them in Lodz are correct.
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Michele Lock
Lak/Lok/Liak/Lock and Kalon/Kolon in Zagare/Joniskis/Gruzdziai, Lithuania
Lak/Lok/Liak/Lock in Plunge/Telsiai in Lithuania
Rabinowitz in Papile, Lithuania and Riga, Latvia
Trisinsky/Trushinsky/Sturisky and Leybman in Dotnuva, Lithuania
Olitsky in Alytus, Suwalki, Poland/Lithuania
Gutman/Goodman in Czestochowa, Poland
Lavine/Lev/Lew in Trenton, New Jersey and Lida/Vilna gub., Belarus