Can't login to iCloud through System Preferences, web is fine

Might be a corrupt .plist.

Do a backup, preferably 2 backups on 2 separate drives.

Quit System Preferences.

Go to Finder and select your user/home folder. With that Finder window as the front window, either select Finder/View/Show View options or go command - J. When the View options opens, check ’Show Library Folder’. That should make your user library folder visible in your user/home folder. Select Library. Then go to Preferences/com.apple.systempreferences.plist. Move the .plist to your desktop.

Restart the computer, open the application, and test. If it works okay, delete the plist from the desktop.

If the application is the same, return the .plist to where you got it from, overwriting the newer one.

Thanks to leonie for some information contained in this.

See this If you can't sign in to your iMessage account or FaceTime on your Mac - Apple Support

Also , read and write permissions if changed in shared , guest , home folder and date and time settings , location services are changed problems will always come in to sign in with web iCloud services ( it will affect in messages , face time , or in iTunes , app store and also in appleid.apple.com , www.icloud.com

, just check out read and write permissions and specially date and time settings .

Try setting up another admin user account to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account

Safe Mode - About

Safe Mode Startup – El Capitan. also Sierra

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor. Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown.

Etrecheck – System Information

See this article About macOS Recovery - Apple Support

macOS Recovery is part of the built-in recovery system of your Mac. You can start up from macOS Recovery and use its utilities to recover from certain software issues or take other actions on your Mac.

Hold down Command-R or one of the other macOS Recovery key combinations on your keyboard immediately after pressing the power button to turn on your Mac, or immediately after your Mac begins to restart. Continue holding until you see the Apple logo or a spinning globe. Startup is complete when you see the utilities window .

Now choose terminal type resetpassword and then click on enter key close the terminal window and then click on save , and restart your Mac .

After entering your admin name and password , open system preferences > iCloud if it prompts for iCloud Keychain verification code ( it could be a six digit ) , if you are having it enter it a sms will come to the trusted mobile number , you have to enter that also .

After that , the steps are very lengthy and complicated , if you are expert in these follow up the upcoming steps ( I have not written all the steps , if I would have them in my pages I will definitely wrote each and every step ) , to be honest I faced the same issue long time back but with the help of iCloud Keychain verification code and all of my Apple ID , iCloud passwords I was able to sign in with Apple ID and password every where , but needed the help of apple support senior advisor .

Note : Every one knows if major problems come in Mac reinstall the os , but iCloud Keychain verification code always help in such scenarios .

its correct , if you sign multiple times with Apple ID with incorrect password , the id could be locked for security reasons , even if you create a new user account it won't help as the Apple ID is locked from the server also deleting com.apple.systempreferences.plist from preferences folder of user library will not help .

You have to contact apple support .

After entering resetpassword in terminal restart the machine , I have set name and password in users and groups , so after restart to prompts to enter admin name and password ( yes the same as you asked user name and password ) , in you case it depends what way you have set . But I doub't do you have a iCloud Keychain verification code with you ? verify the screen shot and compare it with yours are they same .

In these screen shots- open system preferences - click on iCloud the keychain box is checked - if you click on options the trusted mobile number will be shown and your country code also .

click on finder , take cursor on top menu bar > Go > Home

Right click on shared folder > click on get info .

See this article Get help with iCloud - Apple Support

I get an incorrect user name or password alert when attempting to sign in at iCloud.com, in iCloud preferences, in iCloud for Windows, or in a third-party app

• Make sure that you're entering your entire Apple ID when you sign in (for example, if your Apple ID is username@me.com, make sure that you aren't trying to sign in with just “username”).
• Verify that the Caps Lock key is not enabled. All passwords are case sensitive.
• Make sure that you're using the correct iCloud-associated Apple ID and not a different Apple ID used for other services.
• If you use your Apple ID to sign in to third-party apps, make sure that you have two-factor authentication and sign in with an app-specific password. If you don’t have devices that can be updated to iOS 9 or OS X El Capitan, you can use two-step verification instead.
  You can see if you have two-factor authentication or two-step verification from your iOS device. Go to Settings > [your name] > Password & Security. You might need to enter your Apple ID password.
• If needed, reset your Apple ID password. -- note : try to reset the Apple ID password .
• May be Apple ID is locked for security reason try to reset the password after 24 hours see this article If your Apple ID is locked or disabled - Apple Support , try this method otherwise apple support advisors will help you .