As my musical education and practice is from a different note
language, I cannot really say much about the motivation of that
approach. In my country one would never call a "cis" just "c" when
talking about music, not even informally (but then nobody wants to
get caught being informal anyway). Is this really significantly
different in English?
Yes, it is, AFAIK. A similar situation exists in countries where
solmisation is common. If you use do, re, mi, for singing a melody,
accidentals are normally omitted.