I’ve always seen the two as different but like you said it depends on the writer.
In my experience though “immortal” is mainly used in the sense that there’s no physically possible way for a mere human to kill said immortal.
It could also be the difference between something that’s biologically immortal (like lobsters who could theoretically grow forever due to their ability to reconstruct telomeres) and something physically immortal like certain atoms. Or event potentially something “essentially immortal” where by all accounts to human life they will outlive us by eons and there’s nothing we can do to even affect them let alone cause damage.