For me personally, it depends on the source of the errors.
If they are errors of transcription (e.g. by Ancestry, or FindMyPast etc.) I do correct them. I treat the census assistant as a ‘transcription’ of the original document, so try to record each part as close to that as I can get it. But I might also record the change as a note (usually, but not always in the Assistant) in case I need to search for it there again, one day.
If it the error on the original document, then I record it in the assistant fields exactly as found, and then (usually) I will add a note explaining the ‘error”. Not just for names though – same applies for every part of the census. Relationships are frequently wrong (wife when not married, visitor for cousins, etc..), as are dates and / or ages.