To change the caret color,
this must be done on a theme-by-theme basis.

Go to “Settings”->“Style Configurator”.

Your current theme name is displayed at the very top.

In the far left column, select “Global Styles”.
In the next column to the right, select “Caret color”.

Under the heading ‘Colour style’,
select the ‘Foreground’ color that you want the caret to be colored as.
I do not think that the ‘Background’ color has any effect.

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Update - I consider this page an ‘update’
because this page is locked:
“cursor invisible”
https://notepad-plus-plus.org/community/topic/11284/cursor-invisible
and that page mistakenly stated that
“Only width of caret, and blink speed can be modified,”
which is not the case.