If we could find a name, you wouldn't recognize it anyway. The best answer is that pools were probably developed as soon as somebody learned how to collect water in an artificial basin. The answer has been lost in time.
Swimming pool go back at least to Roman times Pre 2,000 years many of them were even heated.
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Purpose built pools for bathing has been around since at least the ancient Romans. Far enough back so that no one can any longer say who built the first one.