Prove that I visited a site for medical school interview
10 days ago...immediately signed up for a time slot on their
  website...how do I prove that I visited the sign-up page on the 12 or
  13th of December?


History

Safari keeps history in a database that's capped at 80M.  That's quite literally a metric ton of browsing you would have to do in the span of 10 days for the history to be over written.  If you're not seeing it in your history, and especially if there are older entries, it wasn't deleted from your history, it just simply isn't there.

There are two more places you can look:


Cookies - ~/Library/Cookies
Cache - ~/Libary/Caches/com.apple.Safari


Cookies

If the site you went to had a form that you were filling out, chances are they were using cookies (if just to identify your session).  Search in your cookies cache directory for anything that potentially matches the site you were on.  Listing this directory will show all the cookies stored in your user profile:

$ ls ~/Library/Cookies


Caches

You can search the Caches to see if anything was stored during that time.  Using a similar method to the link above to search for cookies, create a text file called commands.txt (in your caches directory) with the following:

.mode csv
.once caches.txt
select time_stamp, request_key from cfurl_cache_response where time_stamp BETWEEN date('2018-12-11') AND date('2018-012-14');


Then issue the command:

$ sqlite3 Cache.db &lt; commands.txt


If there's anything in that time period that matches your site, then you have "evidence" that you were there.

Conclusion

I don't see how this can help you in your situation.  This data is useful if a manager was trying to prove that you were somewhere (a website) where/when you shouldn't have been, for example.  However, you being at a particular website at a particular time doesn't negate the fact that the required task wasn't completed.

This is like standing in line for days to be the first to buy the very first iPhone ever made only to not pay for it.  It doesn't matter that you were there and you may even have pictures of you holding it; you didn't do the critical part and actually complete the transaction.