All that matters when it comes to getting a Platinum Medal in Proving Grounds is how long it took to complete it. Total Points doesn't matter at all.
Unfortunately, all that matters for determining whether or not you got a "New Record", and whether or not it overwrites the old record, is how many Total Points you got. (Unless you get a New Record, the run is not saved.)
My first run of Speedy Von Speeds Alot was 15:63 and had 3 kills (6,000 points) for a Total Points of 84,393. My second run was 14:22, but only had 2 kills (4,000 points) for a Total Points of 83,341.
Even though the second run awarded me the Platinum Medal, it didn't overwrite the Gold Medal run of 15:63, because the Platinum run had a lower Total Points.
I finally got a New Record by getting 14:49 with 3 kills for a Total Points of 84,930.
ANY Platinum should overwrite ANY Gold, but it won't if it is fewer points. Just something to watch out for. Don't do too well on a Gold run ;)
Gah, same problem on "Big One"
I killed everything and got all the items and got the Gold Medal. Then on the next run, I killed the bare minimum, only got the necessary items, shaved the time down 2 full minutes and got the Platinum Medal, but I got 20,000 less points, so the Platinum didn't count.
Since I had almost a minute to spare, I just killed the whole last room on the next run, and that put me over the Gold Medal score and got me the final Platinum.
I know Platinum doesn't really matter, it's scores that get posted, but I like seeing all those Platinum Medals ;)