A family of Microsoft word processing software products for creating web, email, and print documents.
I hope I can make this clear enough for you.
"Click to Run" is not a program that you install and run on it's own. It is a "support" environment, separate versions of which Office 2010 Starter and Office 365/ 2016 both run inside of. You don't install click to run, it is installed as part of the process of installing Office.
Office 2010 Starter, as you noted, was a "bait and switch" tool to up-sell users on (at that time) Office 2010. But many users found it adequate to their needs. Which meant that MS thought it was too successful on it's own, and NOT successful in the purpose they intended it for (making $$$ MORE MONEY $$ $). So they very quickly stopped even offering it.
Office 2010 Starter was intended to be factory installed only. MS did not provide the installer the general public. So, in other words, "officially" you can't install it yourself. "Officially" the only way you can get it back is if you use the Windows Recovery Partition to "restore" windows to "factory fresh". It would wipe out all of your existing personal files and applications, but you would have Starter back. Then you would have to upgrade to Windows 10 again (and starter may be dropped in that process!, but I have seen confirmation that it can be run on Win 10 with the appropriate update applied) and re-install all of your personal programs, including Office 2016/365.
If you search the internet you MAY be able to find the 2010 Starter installer file: ‘SetupConsumerC2ROLW.exe’ You have to decide if you can trust that site to not include malware in the setup program. I can't suggest a site, because when I do, MS shuts them down ... <sigh>.
If you find it, I can give you instructions on how to make Starter work on Win 10.
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Your current problem is that Windows default is still pointing to Starter, but the program appears to no longer be installed. First thing, go to control panel > Programs and Features. Look through the list of installed programs to see if 2010 Starter is there. If it is, do an Office Repair of it.
If Starter is not in the list, you should be able to do an Office Repair of Office 2016 to reset the Windows defaults to point to 2016.
REPAIR OFFICE 2016 / 2013 / 365
Here are some similar articles describing how to do Office Repairs for various versions of Office and Windows. Pick the one that easiest for you to understand
How to Start Repair in Pictures Win 7-10 https://support.office.com/en-US/Article/Repair...
Repair Office 2013/2016 in Pictures http://answers.microsoft.com/en-us/office/wiki/...