In Dutch "zee" means "sea" and "meer" means "lake", but in German "das Meer" means "sea" and "der See" means "lake".
Similarly, verbs like to want, to need, to have, to desire, etc. are all mixed up.
Who got it "wrong", and how did that happen?
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Kaleissin is right that the original Proto-Indo-European root *mori-, mōri-, "sea/swamp/lake?", was probably just a large body of non-flowing water, be it technically a sea, swamp, or a lake (but not a river). Cognates in English are mere "sea (obs.), lake/pond (poet./dial.), marsh (dial.)", marsh, and moor, but Latin mare, "sea". For other reflexes see this list at the University of Texas, which is based on Pokorny. Be warned that their website is not always completely accurate.
The article on Meer in Duden's Herkunfstswörterbuch explains it too:
The word sea/zee/See comes from Proto-Germanic *saiwi-z, "sea/swamp/lake?", Gothic **saiwi-*, "sea, marsh, lake"; it is uncertain where this stem came from. Some suggest it might have come from Proto-Indo-European **sai-*, "sorrow, sore" (Pokorny). In any case, it appears there is no clear boundary between sea and lake in this root either.
From Kluge's Etymologisches Wörterbuch der deutschen Sprache:
All of the words you mention are in Carl Darling Buck's "A Dictionary of Selected Synonyms in the Principal Indo-European Languages", from 1949. For instance, "sea" is 1.32 while "lake" is 1.34. (You can get a copy at for instance amazon, but be prepared to use a magnifying glass.)
As it says for "sea": "... between 'sea' and 'lake' there is no rigid demarcation (either by size, or as salt vs. fresh water), and the same word or related group may serve for either or both, or shift its prevailing application with changed physical conditions. This is is notably the case in the Germanic languages, with the divergent distribution of the groups represented by NE sea and NHG meer." Any typos are mine, very small and unclear type in my copy of the book...
Nobody got it "wrong", and it is safe to assume that every word has its own story and its own path through time. That is after all why etymology can be so fascinating.
Mallory and Adams (2006) argue that "consensus is probably still in support of projecting an original meaning of 'inland body of water' that was changed to 'salt water sea' in some language groups, e.g. Celtic, Italic, and Slavic" (p. 130). In support of their claim they point out the following:
in Hittite, marmar(r)a meant 'a body of shallow standing water; swamp";
the Greeks and Indo-Aryans borrowed words for 'sea' from non-IE languages;
in OE, there were both mere and sæ.