These symbols for left or right hand notes in the other staff don't really have a good names, but I've seen them called L-shaped brackets, or LH/RH half brackets in a music context.
In Elaine Gould's Behind Bars - The definitive guide to music notation on p. 305, they are called "incomplete brackets". But this is merely descriptive and not particularly useful if you are trying to search for them.
Gould also mentions the complete "square brackets" which are used for arpeggios that are wider than the spread of both hands. LilyPond calls these complete brackets "arpeggio brackets", as seen in links given in @Richards answer.
Unfortunately the most recent version of LilyPond still doesn't have a standard inbuilt way to print the incomplete L-shaped half brackets.
However, you can use unicode characters:
Top Left Half Bracket (U+2E22) : ⸢
Bottom Left Half Bracket (U+2E24) : ⸤
in place of fingering numbers as demonstrated below. (You'll need to tweak the Y-offset if the notehead appears in a space.)
\version "2.23.6"
\score {
\new PianoStaff {
\set PianoStaff.fingeringOrientations = #'(left)
<<
\new Staff {
\clef "treble"
<c' f'>4
<e'-\finger"⸢">
}
\new Staff {
\clef "bass"
<g -\tweak Y-offset #-0.5 -\finger"⸤">4
<a c'>4
}
>>
}
\layout { }
}
Another option with more control than the unicode characters (and only slightly more code heavy) is to draw these symbols with lines and store them in macros, as demonstrated on Leigh Verlag's blog:
