I have yet to see a Japanese textbook saying katate-uchi is a type of sword and different of uchi-gatana. At least the term katate-uchi gatana would be more correct and Japanese are known to shorten locutions.
"Definitely, the first impression is a popular uchigatana shape to accommodate Katateuchi one hand"
His father was Gorosaemon-no-jo Kiyomitsu, and his active time was around the Tenmon era, and many of his swords were shorter, and the nakago were katateuchi (intended to be used with one hand) or a little longer.
Yosazaemon no jo Sukesada and Gorozaemon no jo Kiyomitsu, their swords are usually 2 shaku to 2 shaku 2 sun which is short, and both the mihaba and kissaki are standard, and there is a short nakago, which is a katateuchi uchigatana shape.
The upper half has sori, and from this, it is possible to judge this as a shape seen often around the date on the ura side date (around the Tensho period) for a katate-uchi or short uchigatana.
From 2012, another typical description:
As an almost correct anwer, some people voted for Katsumitsu
and Tadamitsu. These smiths have suguha work with a bright nioiguchi, but their active period was
around the Bunmei and Daiei periods. Because they were active at a later time than Yosazaemon
no jo Sukesada and Gorozaemon no jo Kiyomitsu, their swords are usually 2 shaku to 2 shaku 2
sun which is short, and both the mihaba and kissaki are standard, and there is a short nakago,
which is a katateuchi uchigatana shape. This katana is much longer and wide, and the shape is
from the second half of the Muromachi period, and it is characteristic late Muromachi work, so you
should pay attention to this type of detail.
Explanation by Hinohara Dai.
It may be that more correct use of the phrase would be "katateuchi uchigatana" ... but they are using it to describe a type of sword and a type of nakago in particular and have used it independently as I used it above.
The salient point is that uchigatana isn't the right word to describe these one handed swords. Maybe it is necessary to append uchigatana to the phrase though.
Anyway I don't invent this stuff out of thin air because I feel like it.