How to state that a complexity bound does not depend on a given parameter size?
The usual convention is to state upfront that your big O notations are with respect to $n$, and that whenever the constant depends on some parameter $m$, then you write $O_m(\cdot)$.

If in another paper you are fine with allowing the constant to also depend on the parameter $m$, then instead you state upfront that your big O notations are with respect to $n$, and the underlying constant could depend on $m$. Then you can use Luke Matheison's suggestions that if in some case your big O doesn't depend on $m$, then you mention that in that case the underlying constant doesn't depend on $m$; but that could be more confusing than the convention in the preceding paragraph.