Thanks for posting the video! While there isn't a specific "new record" form at the moment that can be customized like there is for record details, please know that our team is working towards building this functionality, but does not yet have a release date. In the meantime, you can add fields to the form that shows when you expand a record and then add a linked record:
To do this, hop in Edit mode, click on your linked record field > click on Appearance menu > make sure "Cards" is selected (not "Pills"), and then click on Fields and add any fields you want visible > Publish. Here is a short video showing the process.
For now, it isn't possible to add new linked record values from the Add record button in the top right of your interface page. This is another area in which we are working towards making improvements. To stay on top of all things new at Airtable, please visit our What's new page.