"Godzilla: King of Monsters" is the Burr cut, with only ~45 minutes of the original left. It's embarrassing.
Look for "Gojira" if you want to see the original. For a double feature find "Godzilla Raids Again" (Japanese original, not US theatrical release), the second movie in the Toho franchise. Big G fights Anguirus. The meme scene of Godzilla fighting another big monster at the Japanese Capital Building comes from this movie. It was rushed into release six months after they realized they had a hit in the first one. It's the original "big thing fights big thing" in the Gojira-verse.
(All above available on Amazon.)
I'd recommend the Criterion Collection's blu-ray of the original, also available on Amazon and You are not allowed to view links.
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Login. It includes the original and the Raymond Burr cut with a very insightful commentary track on both (the commentator assumes you watch them in that order). It also has a short documentary called The Unluckiest Dragon, which details the incident with the Lucky Dragon Number 5 and how it influenced the final cut and audience reception of
Gojira. I recommend watching everything in this order: the original cut, the Burr cut, the documentary, the original with commentary, and then finally Burr with commentary.