This is Max from the Shopify Team.
Processing partial payments Online or using Draft orders is not something that is currently possible. There are some workaround options that you could use in a case like this.
I would suggest sending a Draft order specifically for the deposit (1st installment) amount. Then you can refer to this initial order (number and amount) in a new draft order for the full amount. For recording the discount, you might want to add a custom discount that refers to this order number or date. Or add a custom item (with the discounted price) and include notes.
Check out the doc all about Draft Orders if you need a refresher.
I hope this helps with the particular type of payments you need.
I also DESPERATELY need this function. I do custom orders all the time that require half down and half upon completion. It seems to me that all that needs to be done is that the client have an alterable field when they click the link to pay where they can whatever about they want in and pay it, thus kicking it from a draft to an invoice with a balance due. Isn't this possible?? I know I don't understand coding but is seems like an easy fix from where I am sitting and a feature that would be beneficial for a ton of merchants. Just a thought... no, a request, lol. Thanks!