Whilst I was preparing a reply, the above has covered some of the points I was to make, but since I've written it out, I'll put it here in case it has some extra help in it.
The terminal output shown in post #1 indicates that the file:
linuxmint-22.1-cinnamon-64bit.iso
is verified, since the output of the sha256sum command applied to the .iso file matches the alpha-numeric sum specified for that iso in the sha256sum.txt file. That .iso file is good according to the terminal output. In this case it appears the terminal method for verifying the iso was effected in the normal manner, i.e.
Download the iso.
Download the two sha256sum files from the same directory.
Have the iso and both sha256sum files in the one directory.
Run:
sha256sum -b <file.iso>
.
Compare the output of sha256sum command with text in sha256sum.txt
On the matter of the asterisk (*), it is the contents of the sha256sum.txt file that contains the asterisk, as part of its format, but it's not part of any verification command the user runs.
There are two verification processes:
1. to check that the .iso file downloaded has been copied truly and is the same as the .iso file on the server.
2. to check that the sha256sum.txt file is a true copy of that file. For this one needs to import a key with gpg and run a few gpg commands.
The results of the first one has been shown as described above. Often the second one is not done, and found not to be necessary, but opinions differ.
However, the Mint Verify Tool, does evidently require that the gpg key be properly installed on the system, which is the key
@gvisoc mentioned in post #5.
The instructions for importing the key are described here:
https://linuxmint-installation-guide.readthedocs.io/en/latest/verify.html
The problem of "Integrity check failed" in the second GUI image appears to relate to the actual iso being tested. In the image the iso is named as:
linuxmint-22.1-cinnamon-64bit(1).iso
which is not a name that appears in the sha256sum.txt file. In that sha256sum.txt file, the name of the verifiable file is: linuxmint-22.1-cinnamon-64bit.iso, that is, a name without the brackets and their contents in the named iso in that image. In that case it's understandable that the iso file with the brackets in its name would fail to be verified, and the message actually says that the sha256sum file does not contain a sum for an iso image with that name, which is the case.
The "1" simply means it was the first ISO I downloaded to the Downloads Folder...nothing more.
This does not appear to be correct. Normally when a file like: linuxmint-22.1-cinnamon-64bit.iso, is downloaded, it is downloaded with exactly that file name, but when a second download of the same file is run, the second file doesn't overwrite the first file, but it is given another name which includes the brackets and the number 1, and is named: linuxmint-22.1-cinnamon-64bit(1).iso. This numbering differentiates the two downloaded files which are the same. It's a sort of "failsafe" for downloading.This phenomenon was also mentioned by
@gvisoc in different words in post #5.