I used to use $X \perp Y$ to denote that $2$ random variables $(X,Y)$ are mutually independent or orthogonal. However, I have come to realize that r.v. independence and orthogonality are NOT equivalent.
Now my question is: how do we differentiate them in notations?
Both $\perp $ and $\perp\!\!\!\perp$ are used for independence as seen here, and context should make clear what the symbol is intended to mean. However, as you note, $\perp $ is not really the best symbol for independence as it can be construed as denoting orthogonality (uncorrelatedness), which is weaker than independence.
In this sense, $\perp\!\!\!\perp$ is a much more fitting symbol for independence, not only because it is distinct from the typical symbol for orthogonality, but also because it suggests a much stronger notion than orthogonality.