I'm using Mac, so it might be different. But, here's a few things you could try.

On the Menu, Tools, Account Settings: Highlight with your mouse, your Email #2. At the bottom of that column, click on Account Actions. Make sure Email #2 is the Default Account. I think it will check it first, but don't know what order will come after. (Mine usually checks so fast I have no way of seeing which one it's checking first.)

Also, you could try: in the same Dialog box of Accounts. Click on each account, starting with your Email #2, highlight Server Settings, Set Check for New Messages at 10 minutes for Email #2, then 11 minutes for Email #3, then 12 minutes for Email #1.

This probably won't make a difference for manually checking. These are the only settings I could find regarding checking the server.

Email #2 is already set as the default account. It does not check the default account first. It always checks email #1 first, which is the account I added first. This is all noted in my original post.

What I want is to set the checking order for manual checking. That is, when you click on the "Get Messages" button.

I figured out how to do it. When modifying the prefs.js file, it checks accounts in the order of serverN, where N = server #. In general, if you change the server # (throughout the entire file) for each mail account to a numerical ordering that matches the order you want the accounts checked in, it works.

I'm not sure how idN affects it, but there was a corresponding matching identity # for each server #, so I changed the identity # to match also.

It's now checking accounts in the order that I want.

In case anyone else tries this, please remember to quit Thunderbird first and back up your original prefs.js file before attempting (at your own risk). ;)

Wow, that's cool that you figured it out. Nice for you to share and help others, thanks. Did you use a text editor to edit the file? Or a special app?

Yes, I just used a text editor, since it's a text file.