The printer’s manuscript has “all that day”, whereas the 1830 edition has “all the day”. The 1908 RLDS edition restored the reading of the printer’s manuscript, as did the 1981 LDS edition. Both of these corrections were made under the assumption that here the 1830 edition descends from 𝓟 (when in fact it was copied from 𝓞).
The phrase “all that day” is supported by usage elsewhere. In the current text, we get 12 occurrences of “all the day”, but in each instance this phrase is followed by long (“all the day long”). When there is no long, we get occurrences of only “all that day”:
It thus appears that in 3 Nephi 1:12 the 1830 reading, “all the day”, was an error.
There are two examples in his copywork where Oliver Cowdery initially wrote that instead of the correct the, and in both those cases Oliver caught his error:
On the other hand, there are two instances where the 1830 typesetter replaced that with the, probably accidentally:
It should be noted that all of these errors, both Oliver’s and the 1830 typesetter’s, deal with geography (city, land) rather than with time (day). In any event, the error here in 3 Nephi 1:12 could have occurred in either 𝓟 or the 1830 edition. The critical text will therefore follow the systematic usage found elsewhere in the text, “all that day” (in distinction to “all the day long”).
Summary: Maintain in 3 Nephi 1:12 the determiner that in the phrase “all that day”, the reading in 𝓟; here the 1830 typesetter apparently replaced the that with the, thus creating a unique reading for the text (an instance of “all the day” without a following long).