In some contexts it is important to express whether a given date (for example October 25 and November 7 in 1917) is according to the Julian or the Gregorian calendar. Are there established Latin phrases for this? For example, should I say Nonis Aprilibus anno MCMX ad calendarium Iulianum/Gregorianum? Latin was in wider use when both calendars were used at the same time, so I assume there was an idiom for this purpose.
Whatever it is that you intend to write, and if you need to make the distinction, it has obviously to be set at a time after the adoption of the Gregorian calendar. As I mentioned in a comment on the question, the two systems after the change were simply referred to as 'O.S.' (Old Style, i.e. Julian) and 'N.S.' (New Style) in English writings.
In adopting those references, a satisfactory word would be ratio, which I suggest is preferable to either modus or genus. I should myself choose any of these before stilus, although in classical times the word was sometimes actually used with a secondary meaning similar to that sought here — but that's just a personal preference. In the choice of adjectives there are various alternatives — for instance prior and vetus, recens,novus and perhaps infimus, none of which seems controversial — though one might just as comfortably (and more clearly) use Julianus and Gregorianus instead of 'old' and 'new'.
In a later context, in times when it has/had become customary to use simple ordinals for the days of the month in English, it's not really appropriate to use the method of expressing dates from classical times. By the time the Gregorian system had been adopted across Europe, if not earlier, the old Ides of March, for example, would generally be written in correspondence as something like dies XV mensis Martialis A.D.MDCCLXXVII.