I do not know EXACTLY where either slip was/is but they look suspiciously similar just based on a couple of photos - one could wonder if the 2023 incident is a collapse of the 1961 repair ?
Not quite the same place, but in the same swamp!
The 1960 landslip happened on 27th December, and was found by ganger Stan Kimber early that morning. The slip was estimated at 10,000 tones, but it took three times that to rebuilt the embankment. The Aldershot News had several reports (though being a local paper may not be 100% accurate).
The site of that slip was a culverted stream near Totters Lane, which must be the one 200 m to the east of the Whitewater (itself at the field boundary just to the east of the current slip). Constructing a solid foundation in a swamp required piling and a laying bed of ballast. The repair was predicted to take two weeks, but took seven (reopening on 13th February 1961). From 19th January a temporary timetable was introduced recognising the 10 minute delays over the section where one line had been slewed across.
The reports repeat the usual comments about how often the Winchfield-Hook part of the line has suffered slips. There was a recent one on the south side of the line at exactly the same point - that sounds like the one on 16th February 1951. That was at a stream (a "racing torrent" at the time), and about a mile and a half from Winchfield. But while "part of the embankment had dropped some 25 feet", into the boggy ground, that happened on Friday morning and was fixed by Monday evening!
One "colour" story was about a landslip that demolished a bungalow with its occupants in bed inside - at an unspecified date, unfortunately. There is also this, about landslips in cuttings: "This has now been largely overcome by the planting of trees and the afforested banks hold much better".