Error messages and install failures with Linux Lite 5.0

[Şerban S.]

[...] old Toshiba Satellite: [...] The machine does not boot at all though. I just get a flashing cursor in the upper left corner. [...]

Hi!
1. Too few information to be able to figure out at least two scenarios. A hardware dignosis or at least the specific model of Satellite (this is a very generic name, there are lots of Satellite!), would help people here to figure out what might go wrong.
2. Given the scarce info, I can only make assumptions, based on practice. So, here is my guess:

Something went wrong at some mid stage installation, when the installer writes down the GRUB onto the boot sector, then further.
There is only one approach I can figure out right now: do you have at least 1 USB port?
If you have, then write a small distro called "BootRepairCD" on the stick, then run the App.
It will run a diagnostic tool and do a bootloader restore, if it finds anything.
You can use a CD, but it's very slow and maybe, might cause some trouble. Or... maybe it is the source of all trouble...
Further more, many old CD devices, give read errors without any clue to figure out how to fix this.
Maybe a glitch during installation lead to a failure that was somehow skipped and the installer moved on without signaling anything, for whatever reason. This usually leads to incomplete installs. Fixing such an installation is difficult and requires a deep understanding of a Linux distro and of the processes from boot to startup.
If the computer is so old that has only an optical device for this kind of purpose, I'd rather buy a SH computer. Here, in Romania, a good  SH computer, goes at about 200 -300 euros (Core2 Duo, HT, 4 GB RAM, 320 - 500 GB HDD), but you can find a cheaper solution, if on a shoe-string budget.
I have an Lenovo ThinkPad T400 that seems that is living its last days, but we had a long journey together and I was extremely pleased to work with it.
I now have to move to something else so maybe you should consider a small hardware upgrade.

I hope this helps.
Maybe we can come with better ideas if you post the full hardware configuration of the machine.
All the best!