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Minor correction: they are derived, not influenced by Vulgar Latin.
That's why so many words are different from Classical Latin, but similar between Romance languages. Like how Latin for house is "domus", but Romance languages use casa/casă/chez because common people referred to their house by the word "casa".
https://en.wiktionary.org/wiki/Reconstruction:Proto-Slavic/x...
https://en.wiktionary.org/wiki/Reconstruction:Proto-Slavic/p...
https://en.wiktionary.org/wiki/Reconstruction:Proto-Slavic/j...
https://en.wiktionary.org/wiki/Reconstruction:Proto-Slavic/b...
https://en.wiktionary.org/wiki/Reconstruction:Proto-Slavic/g...
The phrasing "evolved in the past due to the influence of" has a more expansive meaning than "influenced by."