Here's how you can customize the welcome Email and text message.
Go to Clients > Notifications:
Scroll down to New Client, there are two separate notifications: for new clients who sign up on your website as well as for new clients who get an Invite from you. If you want them all to have the same experience update both settings to the same text.
To further deliver an exceptional onboarding experience consider creating a hidden (no menu link) web page and including a link to that web page in the welcome text. You could include more information for new clients and provide them with a link to start booking specific