You are absolutely right about the first part: invite e-mail is sent basically to prompt the customer to complete their registration by setting their password.
About the welcome e-mail: it is sent only after the created customer gets activated. So if a customer makes an order without activating their account, then they will not get a welcome e-mail. Basically, it's an e-mail that is sent after they create an account and activate it by assigning a password to it.
It may be helpful to understand different customer account 'states' that Shopify sets, as well as the steps a customer (or you) has to take to get there:
Customer states set by Shopify
Post-purchase or Newsletter subscribers
A customer record will be created for anyone who purchases a product from your site, or for anyone who subscribes to your email newsletter. Both examples result in an account status of "disabled" - the default customer state. Basically, you have a record of these customers but they do not have an account.
You can invite these customers to create an account and their status would change to "invited".
Customer account registration
By default on Shopify when a customer completes your registration form they will automatically be granted access to an account (status = "enabled") even if they used a bogus/incorrect email during registration.
Unique customer flows
Many businesses rely on customer flows that are not possible natively on Shopify. The Customer Fields app enables unique customer flows not normally possible on Shopify as well as some new account statuses (pending, denied):
1. Require email verification prior to account creation
Using the 'email verification' option forces a customer to check their email and click a link to activate their account. If a customer enters a typo in their email address (or attempts to sign up with a bogus email) they will not receive the verification email and cannot gain access to a customer account. This method also reduces friction from the initial sign up process because the customer doesn't have to set a password. The customer will be prompted to create a password once they receive the 'account invite' email triggered by completing your registration form.
2. Require account approval
If you need to collect more information from customers before granting access to an account, requiring admin approval is ideal. For example:
We sincerely hope this added functionality to the customer registration process adds significant value to those businesses which depend on them! Learn more about account statuses and Customer Fields.