Create Credential: Where is the Smart Card Certificate?

Using Edge/ IE11 (no OEMs) to login to a trusted government resource on a machine where the OP is the sole user. The requirement is to type username and password every time the site is loaded as for some reason there is no credential record created for it. Now to add a certificate based credential, we try:

Control Panel\All Control Panel Items\Credential Manager\Add a Certificate-Based Credential

But it wants a smart card certificate. Never had one or a reader. But the invisible certificate is somehow classified as smart card.

Also tried the url of the resource to add a Windows Credential thus:

Control Panel\All Control Panel Items\Credential Manager\Add a Windows Credential

But after filling in the details get:

The information cannot be saved. Make sure the information is correct and that all required fields are completed. Error 0x80070057 the parameter is incorrect

On another machine, there is no need to view the same smart card certificate as the credential record exists and the details are passed to the browser.

Can the credential be created at all?

Having checked the "other" machine for the existence of any smart card certificates for the above mentioned site and found none, also came the epiphany that as Chrome was the default browser on that machine, Google Passwords does not interrogate the Windows Credential DB, but has its own repository.
This is in tune with what grawity has suggested above, and will accordingly update this answer if there is some other way to include the logon details for the site as a Windows Credential, without the accompanying invalid parameter error.