Hi
According to your post:
However, Wix also send a very ugly looking plain-text email with title: “You just purchased a plan” (and this email also includes my personal email address at the bottom, not an email from my domain, which is non-profesional).
You have asked the following question:
How can I disable this automatic email being sent
The emails which are sent out when someone subscribes to a plan through Wix Paid Plans App is not customizable. You can see the article provided by Wix Support here: https://support.wix.com/en/article/request-sending-a-customized-email-to-paid-plan-customers-after-purchase
Also as @givemeawhisky said above this is not a code issue and Wix Support will be able to help you best.