That still is the case, now.
It is this nomenclature which gives rise to the term ‘bisextile year’ found in the Book of Common Prayer. The curious 48-hour day was eventually recognised as two separate days but, importantly, it was the first of the two that was considered to be the intercalary day. A modern pedant might therefore write the dates of the last week of February in a leap year as:
23, 24′, 24, 25, 26, 27, 28
Modern practice might be considered merely a relabelling:
23, 24, 25, 26, 27, 28, 29
This relabelling does not change the fact that 24 February is strictly the extra day if not the extra date.
It seems unlikely that many people are greatly troubled by the above analysis but in the late 1990s both Sweden and Finland formally enacted legislation to change the intercalary day from 24 February to 29 February.
One of the few manifestations of this change relates to the Saint’s day for St Matthias. Finnish almanacs for 1996 show his Saint’s day transferred from the usual 24 February to 25 February on account of the leap year but almanacs for 2000 show his Saint’s day retained at 24 February. The Roman Catholic Church ducked this problem by moving the Saint’s day to 14 May but that was to avoid occasionally having to celebrate St Matthias’s Day in Lent!
While most designers would consider that the right place to insert this short day is between 28 February and 1 March, super pedants should insert this short day between 23 February and 24 February.
The Julian calendar reform defined every fourth year to insert a day
ante diem bis sextum kalendas martias.
That is 6 days before the kalends of March, or as we like to count it: the 24th of February. 24th February is –the– leap day, as it is to be counted twice. A bit impractical for our primitive incrementing reading of how our calendar seems to work?
Only that the named days afterwards in February get a +1 number, so that from 25–28/9 everything is shifted. Since the Catholic church is also the Roman Catholic Church, we see that at least until 1970 (there is some theoretical debate around this, in perhaps a bit of contrast to this otherwise fine answer) this is also reflected clearly in their Name days. They are in older prayer books listed two times for 24th February!
Now it reads:
Die 24. februarii, Sexto Kalendas martii
Die 25. Februarii, Quinto Kalendas martii; vel in anno bissextili: Sexto Kalendas martii
To really get down to the construction, highlighting some of the apparent advantages and disadvantages of either method, it reads as:
The Church of Rome has adhered to the ancient Ecclesiastical Calendar, in placing the intercalary day between the 23rd and 24th of February, as the old Julian Calendar did (Art. 17); thus making, in fact, two 24ths, as in the Julian Calendar there were two vi. Kalend. Hence the change of Sunday Letter, in Leap-year, takes place in the Roman Calendar after the 24th of February; not, as with us, after the 29th. The note in the Roman Breviary and Missal is
"In anno bissextili, Februarius est dierum 29; et Festum S. Mathiso celebratur 25 Februarii, et bis dicitur Sexto Calendas, id est, die 24 et die 25, et Litera Dominicalis quae assumpta fuit in mense Januario mutatur in pnecedentem : ut si in Januario Litera Dominicalis fuerit A, mutatur in praecedentem, quse est g, et litera f bis servit, 24, 25."
The same rule is laid down in the Sarum Missal:
"Si Bissextus fuerit, quarta die a Cathedra S. Petri (February 22) fiat Festum S. Mathiao."
In our first Prayer Books, the ancient rule was followed, or intended to be followed, as to the mode of intercalation, and the change of St. Matthias’ day to the 25th in Leap-years. In the Church Calendar, the place of the Refugium was taken by St. Matthias’ day. But at the final revision, in 1662, the ancient practice was given up, and the Civil mode of intercalation — namely, making the 29th day of February the intercalary day — was adopted. The Revisers of 1662 were, probably, induced to make this change in order to put an end to the doubts and mistakes which had arisen respecting the proper day on which the Festival of St. Matthias should be kept (‘).[…]
Thus, in both modes of reckoning, and whether the year be Common or Bissextile, March 1 has always the Letter D. In the Roman Calendar, the intercalation does not disturb the arrangement of the Calendar Letters, except from February 26 to February 28. In the Civil Calendar (that of the Anglican Church) there is no displacement at all of the Letters. February 29 merely takes the same Letter (D) as the following day, March 1.
— Samuel Butcher: "The Ecclesiastical Calendar: Its Theory and Construction", Hodges, Foster and Figgis, 1877, p27. (archive.org) [Note that it talks about adapting 'civil' and 'Anglican church' intercalation, not Catholic!]
Nihil Nove Sub Sole: What does "leap-year" mean? Where does the term “twice sixth year” come from?
The authorities surrounding this being Julius Caesar and the Catholic Church, and that it is still the leap day, I'd like to answer the
Q: Was leap day ever officially observed on February 24?
— C. Philipp E. Nothaft: "Scandalous Error: Calendar Reform and Calendrical Astronomy in Medieval Europe", Oxford University Press: Oxford, New York, 2018.
Which day is the leap day?
The Gregorian calendar is a modification of the Julian calendar first used by the Romans. The Roman calendar originated as a lunar calendar (though from the 5th century BC it no longer followed the real moon) and named its days after three of the phases of the moon: the new moon (calends, hence "calendar"), the first quarter (nones) and the full moon (ides). Days were counted down (inclusively) to the next named day, so 24 February was ante diem sextum calendas martii ("the sixth day before the calends of March").
Since 45 BC, February in a leap year had two days called "the sixth day before the calends of March". The extra day was originally the second of these, but since the third century it was the first. Hence the term bissextile day for 24 February in a bissextile year.
Where this custom is followed, anniversaries after the inserted day are moved in leap years. For example, the former feast day of Saint Matthias, 24 February in ordinary years, would be 25 February in leap years.
This historical nicety is, however, in the process of being discarded: The European Union declared that, starting in 2000, 29 February rather than 24 February would be leap day, and the Roman Catholic Church also now uses 29 February as leap day. The only tangible difference is felt in countries which celebrate 'name days'.
— Academic Kids: Leap Year
In summary, we see that from the Julian reform to the late Middle Ages, the 24th of February was the undisputed leap day. Ever since then this date is very slowly giving slowly way to the 29th of being recognised as 'it'.