Correct, the "send free link" doesn't simply issue an identical thank-you message to what would be issued after an actual purchase. Instead, it sends the link with whatever message you enter in the "Send free thank-you email/download link" screen itself, so you can tailor that message to your purpose for sending the free link.
We do not simply issue the product's or account's standard post-purchase thank-you email because an actual purchase/payment would likely not be involved in issuing a free link, so any verbiage having to do with purchase/payment could be confusing to the recipient, and any template variables having to do with actual order data would have no values to substitute.
BTW, note that any Common Thank-you Email message you specify in Admin > Account Prefs would be sent ~in addition to~ any product-specific thank-you emails, unless you configure each product to not send its own email. Tangible goods do not issue individual-product emails if you don't type anything in the Email Message field of their product configs, and digital goods would not issue individual emails if that field is left blank AND Enable Templated Email is checked.
To customize the appearance of our thank-you pages, aside from adding your own HTML, you can also override the default page styling by inserting some CSS (probably best in the Common Thank-you Page HTML field, so it affects all your thank-you pages), either inline with your HTML tags or perhaps in a block like this:
YOUR CUSTOM CSS GOES HERE
Here's a good tutorial/reference for CSS coding: