My experience is quite the opposite, colmx. People I know in Pattaya do refer to it as a Baht Bus in addition to a Songtaew; never heard anyone call it a taxi.
But who calls it a Baht Bus? A Thai/Khmer/Laos person or a farang? I have never ever heard any Asian call a Songtaew a Baht Bus in 17 years visiting Pattaya... I have only heard it being called that on message boards