Transferring email from one (work) Gmail account to another (home).

I am confused because it sounds like both email accounts are business (gsuite) and not personal (somename@gmail.com).

You said your second account was
username:  my.name@domainname.org
password: password2

That indicates a GSuite account.  If it is your own GSuite account and you are the admin, there is a migrate option.  How long did you wait once you upgraded?  Sometimes it takes a day for the Google info to fully propagate.

I am so sorry. I copied and pasted. That's what happens at 1am, I suppose. i will make the change.

No, I was right. I didn't say that was my second account. I have tried to alter the wording to make it clearer. Those login details with password2 are the details used in the settings of the new personal gmail account for fetching email from the pop server for the source email account. I was just trying to show that the login details were the same email address but a different password that was provided by the company. The two different login details that I gave were for accessing the same mail server: once as someone viewing the email and second as another new gmail account importing the email from that source server.

I waited a day.

https://www.experts-exchange.com/questions/29158183/Backloading-emails-into-gmail.html provides several ways to do this.

Best to stop + take a breath before proceeding + read the following carefully

Doing a mail transfer like you mentioned, then having the transfer interrupt (error out), can be super complex, because now you have one full copy of all your messages (original account) then another copy (new account) with some messages missing, also the last message transferred (when error occurred) by be corrupt.

At this point, you have 2x general choices.

1) Destroy all messages in your new account + start over using one of the approaches mentioned in the above EE question link.

2) Repeatedly run imapsync till all message show as transferred correctly.

Note: You'll have to look carefully at the imapsync source code + maybe open a developer ticket to determine if imapsync is smart enough to determine the one message which might be corrupt should be replaced.

Tip: If possible, for messages with large attachments (like .pdf files) consider downloading the attachments locally, then delete the messages from your email. Many times large attachments are difficult to transfer correctly between various mail accounts.

This is still not clear.

Let's go back and take this step by step.

Despite the fact that I provide authentication details on the domainname.org website, once I am authenticated, I am diverted to a Gmail account with the company logo in the top right corner.

This indicates a GSuite account.

I would now like to transfer all of the emails from this domainname.org (Gmail) account to another new, personal Gmail account.

From this, I assume you mean from your GSuite account username@domainname.org to a free gmail account, username@gmail.com that you later paid for additional space.

The authentication details that I then provided in order to fetch the mail from the pop server were:
username:  my.name@domainname.org
password: password2 [This is a random-looking password provided to me by the company]
pop server: pop.gmail.com

Well, it would have been better to use imap instead of pop. The reason is you may have not preserved email on the original server. Using pop, when an email is imported, it is possible the import process may take it off the original pop server.

I set up another personal Gmail account and attempted to initiate another "Import mail and contacts" process into that Gmail account. It would not even begin. It just said that there was no mail
to fetch.

As I said, it is possible that the mail was deleted from the original server as it was imported. However, I think it is also possible that the email was just marked as already seen which prevented the download.

the "Less secure app access" is "managed by your domain administrator".

This indicates you may be using outlook desktop app. In order to use outlook, the account does need to be set up to allow less secure apps. But you don't need to use outlook for the import process and if you are, that may be part of the issue.

First thing, check to make sure your old company email is still there.  In a new private browser (In chrome it is using incognito) go to https://gmail.com. The reason for private browsing is to allow you to use multiple accounts like username@privatedomain.com and username@gmail.com. You can just as easily use separate browsers such as Chrome and Firefox.

In your new private browser, go to gmail.com. Log in with your company credentials my.name@domainname.org.  Search around and verify your email and contacts are actually there as you expect.  If it is, instead of importing email through outlook, you can do this in your personal gmail account. Open a new private browser and go to gmail. This time, log in to your personal gmail account yourname@gmail.com. You can do this while your other private browser is open. Just make sure you crate a new private browser and not a new tab in the previous browser.  

Once you are in your personal gmail, click on the bigger gear at the top right just below your avatar. Then choose settings. The third tab on the top of the settings area is Accounts and Import. You will have to use POP here, give it a try.  

Another option is to use outlook and imap.  In order to use this method, the company email would have to have less secure apps turned on. More than likely they do as many people use outlook.  Now in outlook, create two accounts both using IMAP. One account will be using your my.name@domainname.org credentials and the other your personal username@gmail.com.  Once you have both of those set up, you can simply drag your inbox (and folders) from your domainname.org account to your personal gmail.com account.  Again, this assumes that there is email in your domainname.org account and less secure apps are turned on.

My 2cents, I would hold off on deleting anything. You don't know if that email is still available or not. Using outlook desktop app, you can save the current emails locally in a new folder. Then delete them from the server. Otherwise, just deleting could cause you to loose those emails forever.

Scott, you have appeared to understand what is going on here....except that I have never used an email client with this company email account.
Also, I have already used the personal gmail account to try to download the email using the "Import mail and contacts" process. I did it when it managed to download two thirds of the email...and then it wouldn't carry out the import again.....and so then I created a new gmail account and tried to do it again. It would not even start.
The company email is still all there.

Seeing as I have tried to transfer the emails from the company account to the first gmail account and then the second, that seems to suggest that the problem is with the company gmail account (the source). You wrote:
"As I said, it is possible that the mail was deleted from the original server as it was imported. However, I think it is also possible that the email was just marked as already seen which prevented the download." How is the email marked as seen? Is it marked in the company (source) gmail or is it marked in the personal (destination) gmail?

OK, I found a solution. According to the Gmail help pages, when an email is POPed, it is then marked as downloaded. It cannot then be downloaded again. I don't really get the normal mode and recent mode thing entirely.
https://support.google.com/a/answer/6089246?hl=en
Anyway, my POP settings in my original Gmail account were "Enable POP for all mail". I assumed that was an abbreviation for the top choice of "Enable POP for all mail (even mail that's already been downloaded)". It may well be. However, I discovered that if I then selected that option of "Enable POP for all mail (even mail that's already been downloaded)" and then clicked on "Save Changes" that it then cleared the POPed status from all the emails.
Then I was able to use a fresh Gmail account and POP all the emails (approx 3000) but it did not download about 200. I am not yet certain of why it chooses not to download some emails but I have seen that it does not download emails containing login details. Maybe the draft emails are counted in ALL MAIL in the source email but are not downloaded to the destination email.

Thank you very much Scott and David. Your help has been very much appreciated.

I have done some migrations in this manner and with a paid migration tool. I ran into the same thing where a very tiny percentage did not come through.  I was disappointed in the paid product other than it had a log of what didn't transfer. For those items, I  just did it manually or ignored.  The only time I was ever able to get everything is a long time ago when I used to host mail on my web serve and could literally move things around or use the admin panel. As soon as Google came out with the Google Apps Beta for free, I converted all of my clients to that and then removed hosting any email. It is too much of a PITA.