control variables and multicollinearity in regression

Dear Team,

reading through the link you added at the section (https://learn.365datascience.com/courses/machine-learning-in-python/a-note-on-multicollinearity/) and doing some further reading, i understand that deciding whether a variable is a control variable or not can be sometimes pretty subjective.
like in the given practice data 'Registration' could be a control or a main variable.

but there is a different rule when you decide that the given feature is a control variable as you can neglect multicollinearity (if it was not binary).

as i understand for the algorithm it doesnt really matter, the math is the same and it is mostly a theoretical distinction.

am i right in that simplified view point and/or  what is the fine line between being control or not.

regards,
peter