System problem detected: solution from Q681231 didn't help

I suggest that you do the following:

delete the crash file(s)
perform a full package update (sudo apt update && sudo apt full-upgrade)
reboot

If you then again get the message "System program detected", then please proceed to report the problem by pressing the related button on the message.

You most probably will receive a bug number, please note it down it for the next steps.

Manfred:
Thanks for answering. When I ran the full upgrade command, during the process, a screen showed. I'm including a link for you to see.

https://drive.google.com/file/d/1F2cFiZf8Iq59okpjuIeUwMKjGEW6dM-d/view?usp=sharing

I didn't know exactly what to do next so I I chose to see the difference between both version, as shown below.

https://drive.google.com/file/d/1fpYDKpFuSjllbg18KdR1tVTMvK_YvFGm/view?usp=sharing

After I clicked OK, it went back to the previous screen. And then I OK'ed to keep the local version. Then I was able to reboot to write this answer. Please, advice.
PS:
Let me know if you can't see the images. Thanks.

What you see is nothing extraordinary.

Some packages install a config file with default values. If you have changed these values, at an upgrade of the package you are asked exactly these questions: do you want to keep the config file modified by you, or do you want the "vanilla" version that comes with the new version.

Usually either solution is ok.

Manfred:
Thanks for clarifying. Well, the problem isn't showing yet, so I assume the problem is solved. Thanks.

Thanks Manfred Hampl, that solved my question.

Manfred:

Aaahhh! The problem showed up again last night. Exactly the same one, as shown.

williepabon@williepabon-Macmini:~$ ls -l /var/crash/
total 21132
-rw-r----- 1 gdm whoopsie 21636451 Feb 29 22:28 _usr_bin_gnome-shell.121.crash
williepabon@williepabon-Macmini:~$

Wonder if you have any other venues to explore. Thanks.

Which button did you press in the window that tells that there was a problem?

Manfred:

This morning, after I booted up, I got the familiar message in the screen abt System problem detected, with the option to report the problem (which I did). Then, I did the ls -l command to verify that it was a recent one.

When reporting the problem, did you get a bug number?

Manfred:
No. I just click the button on the screen message to report the problem.

Does the system work despite that "System error" message?
Is you system fully updated?
What is the output of the commands

uname -a
lsb_release -crid
apt-cache policy gnome-shell
ls -l /var/crash
dmesg | grep -i gnome

Manfred:

Thanks for answering. Yes, the system seems to be working OK. It's just a nuisance, to have the screen message and to erase the crash report from /var/crash/. Yes, as you suggested on this problem, I ran a full upgrade to the system. The results of the commands.

williepabon@williepabon-Macmini:~$ uname -a
Linux williepabon-Macmini 5.3.0-40-generic #32~18.04.1-Ubuntu SMP Mon Feb 3 14:05:59 UTC 2020 x86_64 x86_64 x86_64 GNU/Linux
williepabon@williepabon-Macmini:~$
williepabon@williepabon-Macmini:~$ lsb_release -crid
Distributor ID: Ubuntu
Description: Ubuntu 18.04.4 LTS
Release: 18.04
Codename: bionic
williepabon@williepabon-Macmini:~$
williepabon@williepabon-Macmini:~$ ls -l /var/crash
total 21132
-rw-r----- 1 gdm whoopsie 21636451 Feb 29 22:28 _usr_bin_gnome-shell.121.crash
williepabon@williepabon-Macmini:~$
williepabon@williepabon-Macmini:~$ dmesg | grep -i gnome

It seems that you missed showing the output of
 apt-cache policy gnome-shell

And for
 dmesg | grep -i gnome
is there empty output?

Manfred:
Oohh!. I'm terribly sorry. I missed it. Here is the output.

williepabon@williepabon-Macmini:~$ apt-cache policy gnome-shell
gnome-shell:
  Installed: 3.28.4-0ubuntu18.04.3
  Candidate: 3.28.4-0ubuntu18.04.3
  Version table:
 *** 3.28.4-0ubuntu18.04.3 500
        500 http://archive.ubuntu.com/ubuntu bionic-updates/main amd64 Packages
        100 /var/lib/dpkg/status
     3.28.3+git20190124-0ubuntu18.04.2 500
        500 http://archive.ubuntu.com/ubuntu bionic-security/main amd64 Packages
     3.28.1-0ubuntu2 500
        500 http://archive.ubuntu.com/ubuntu bionic/main amd64 Packages
williepabon@williepabon-Macmini:~$

Yes. The dmesg command produced and empty output.

Manfred:
Thanks. I will follow your suggestion. When the problem shows again, I will get into the details and will copy the information about any bug that shows. How do I communicate this info to you?

Thanks Manfred Hampl, that solved my question.