If a customer makes a partial payment, is there a way to send an updated invoice rather than the original? Is there a way for the receipt to show the reference number and payment date?

Yes to both of your questions.

When you as the coach records a payment:

When you click on CONTACT TAB > CLIENT'S NAME > INVOICES you'll see a summary of all invoices created for the selected client. It is here that you can click on "VIEW" to see the details of a single invoice and record a partial payment. When you record a partial payment, then Invoice Summary Screen will show the invoice with the partial payment that was made and a second line for that same invoice with the outstanding amount still due.

When the client makes a payment:

If a client makes a partial payment while logged into the Console, the original invoice is updated automatically with the partial payment details (including the reference number and payment date) and the outstanding amount. 

For either scenario:

You can re-send the partially paid invoice to the client (as a reminder to pay it fully) by simply clicking on "Resend" from the options drop down list to the right of the particular invoice.