On my system - Linux Mint Cinnamon, hiDPI - the caret in Writer is thin and consequently (at least given its colour) hard to see. My eyes could be better but I suspect many people will have the same problem. There is a GTK hack that affects the caret system-wide, but I do not wish to resort to that because I find the caret to be fine in all the other applications I use.
My cate in SublimeText: https://imgur.com/a/HRPCEO9
My caret in my terminaL: https://imgur.com/a/iQDSKNK
My - anemic! - caret in Writer: https://imgur.com/a/BI5rbkM
Cf. bug #105415.
From UX POV I'd say yes to have the caret sized as on the desktop environment. If there is no default we should adjust the hairline width to a few pixels.
(Cannot confirm as my display has standard dpi.)
*** This bug has been marked as a duplicate of bug 101379 ***