Dear Yvonne,
For an external user such as the personal account ******@outlook.com, you can only use the Anyone link. The internal user share link and the specific user share link don't support the external users even though they have been added as the guests in your tenant.

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When they submit the form, they can click on Save my response to edit to edit the response. But if they haven't logged their accounts in, they will be asked to log in to edit the response.
After logging in, they will see the following.

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Then they can on View in Forms to see the filled and edited responses.

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But you can only see that the form is responded by Anonymous at your side.
Please let the external users in your environment check whether they have logged in the correct accounts to edit the responses.
Thank you for your time and effort.
Sincerely
Cliff | Microsoft Community Moderator