This looks like the Customer account invite email found under Settings > Notifications in the admin, though I see that the message is customized to mention rewards points. Are you using one of the customer loyalty apps alongside this notification to offer rewards?
You can invite individual customers to create an account (stores on the Shopify Plus plan can also send account invites to customers in bulk), or customers can create an account on their own when visiting your store. Either way, the Customer account invite email will automatically include a unique link for each customer, so there's no action required to edit the link. However, if you are using an app in conjunction with this to offer the rewards points, it may be helpful to check in with the app developers about this as well.
The Customer account invite email is not sent to new email subscribers. This email is either sent automatically to customers who created an account on their own, or sent manually to customers whom you've invited to create an account.
Customers who subscribe to your email newsletter or opt-in to email marketing at checkout will instead receive the Confirmation email if you have double opt-in enabled. Below is a screenshot from Settings > Notifications in my test store to show where these two templates live. The Customer account invite email is under the Customer section, and Confirmation email is under Email marketing.
If you would prefer to automatically send a welcome email to all new email subscribers that includes an account invite, I recommend looking into the email marketing apps available in the Shopify App Store.