I don't think it has anything to do with your Javascript or whether or not you are using the new designer or the classic designer.  I think this has to do with how your staff are accessing the process.

If you have made the form process available for public (unauthenticated) submissions, then those submissions will show that they were initiated by Anonymous.

It sounds like your staff are submitting the form through the public submission, and it hasn't authenticated them into the Forms environment.  It sounds like you are directing them to do that because of an issue with the ReCAPTCHA.

One solution would be to get the ReCAPTCHA configured so that it works on your internal server as well as on the public server.  That way your staff can submit the form via the internal server instead of the public version.  This is my recommendation.

Another solution would be to have your staff authenticate into the public Forms environment before accessing the form.  For example, let's say that the URL for the form is https://internalserver/forms/myform when accessed internally and https://externalserver/forms/myform when accessed externally.  Do not direct your staff to https://externalserver/forms/myform to complete the form as it'll just be a public Anonymous submission.  Rather, direct your staff to log in to the external forms server at https://externalserver/forms.  Since that is the Forms environment instead of the specific public form, the system will have them authenticate before proceeding.  Once they have authenticated, then they can access and submit the form (either via the URL or the Start Process page for example).