From what I understand, for example, if I perceive the Tonic note as "C" instead of "G" in a G major song such that I would sing "Do" on the note "C" instead of the note "G", without transposing the song itself, then I would be using Fixed Do Solfege. So the only difference I can tell between the 2 Solfege systems is if I decide to shift the Tonic in accordance with the key of the song or not.
So in other words, learning Movable Do automatically teaches you to also sing in Fixed Do since it's just Fixed Do with the extra step of perceiving Tonal centers, but learning Fixed Do doesn't automatically teach you to use Movable Do. Is this statement correct?