Hello!
As they say, "Whatever works best for you".
I believe when you download an update to GRUB in Debian, it will automatically detect and add Linux Lite. Problem is, a recent LL update also updates GRUB, which will switch you back to the LL boot loader.
I run Lubuntu on this machine as well. The boot loaders bounce back and forth between Lubuntu and LL, depending on which one was updated last (usually Lubuntu).
I finally assigned a different GRUB background for each of them. That way, I know which one is 'calling the shots'...
73 DE N4RPS
Rob