Sharepoint - Open SharePoint Online web with SharePoint Designer gives server error saying it has an old version?

There may be a couple of possible issues and solutions here.

Solution 1

I think that SharePoint Designer (SPD) 2013 is getting confused with your Office versions after and office update. I presume the possible solution would be not to change the registry.

SPD2010 and 2013 on the same machine shouldn't interfere with each other.

What you would need to do is uninstall SPD2013 and reinstall SPD2013. This will re-associate to the correct version of Office that it requires.
It could be that an update messed up the office dependency where it's getting confused with the multiple versions of Office after update (even though it shouldn't).

Solution 2

I presume you're getting this error when connecting to the site, and this issue may have something to do with your account. Could you try and connect with a different user and password?

Go to SPD2013 > Account > Switch Account and enter username and password and restart SPD.

IMPORTANT!

• You should add an account from the Tenant you are connecting to.
  • If you are connecting to multiple SharePoint Online (SPOnline) tenants, you'll have to have at least one account associated in SPD from every tenant.
• Don't forget to restart SPD after adding the new account.

Using IE, I logged into the site with my windows credentials. Keeping IE open, I re-opened SharePoint Designer and then was then able to access the site.

I've noticed this behavior seems to exist if you use multiple authentication providers (i.e. forms authentication with windows authentication).

Could you please try the following steps to see if it works? But this worked for me.

1. Open SharePoint Designer 2013 then click Account.
2. Sign out and sign in again with the right account to access the SharePoint site.
3. Restart SharePoint Designer 2013.
4. Open the SharePoint site again