I see some inefficiencies (and some risk) in your plan to arrive in London, spend 2 days there, then take a train to Glasgow where you'll rent your car.
Your Scotland trip looks like it doesn't really begin until "Day 4" of your trip. Then at the end of your trip, you appear to be planning to take the train from Edinburgh to London, then get on a plane and fly home. Maybe I'm misreading your plan, but I would not either begin or end my trip that way.
You are going to arrive in London pretty tired, yet you're going to stop right there, settle in and stay two nights. If it were me, I'd push on all the way to your actual destination, which seems to be Glasgow. You mention it's "much cheaper" to fly to London, but given how expensive hotels are in London (and everything else in London) I'd wonder if the financial cost really favors arrival in London so much - I'd get out a calculator and make a fair comparison, factoring in actual costs for the "cheaper" flight plus other expenses you will incur. Regardless of any minor difference in cost, I'd consider the even greater "cost" in TIME for going that way. You will be spending your arrival day plus the next day in London. That's fine if you really want that time there, but your trip is quite short, and I'd be looking very hard at how my days are allocated.
Are you really planning to wake up on "Day 14" in Edinburgh, take the train all the way to London, then fly home, all on that same day? I would not recommend doing that, there are just too many things that could cause you to miss your flight home. My rule-of-thumb is to always spend my last night at (or at least not terribly far from) my departure airport.
Instead, consider re-arranging the start and finish of your UK time: First, take a hard look at the actual cost differences between flying in to London, or flying in a city in Scotland (Edinburgh or Glasgow). Assuming you stick with your flight to London, rather than stopping there for 2 days, upon arrival push straight on to Glasgow, either by train or by a separate flight (if flying, you would be on two separate tickets, so you would want to allow at least several hours between scheduled arrival time in London and the departing flight to Glasgow). Maybe take the train as it would probably allow you to snooze more comfortably (though the flight would get you there quicker...). Sleep your first night in the UK not in London, but in Scotland, next morning your trip actually begins (on what you're calling "Day 2" rather than your "Day 4"). On your arrival day (wherever you choose to stop moving), you will be tired, jetlagged, etc so you're not getting much out of that day anyway; you might as well use that arrival day to get all the way to where you actually want to be...for my first place to sleep, I always look at "pushing on through" to my real destination rather than just stopping where the long flight happens to drop you.
If you want time in London, do London at the end of your trip, not at the start. It's more efficient to spend time in any place as a single block (as many days as you choose) rather than splitting it up into some days at the start, some days at the end. You should spend your last night there anyway (to ensure you don't miss your flight home).
Rearranging things this way would effectively "buy" you at least one extra day in Scotland, maybe as much as 2-3 days. In a trip as short as yours, given all the things to see/do in Scotland, that's what I'd do. Hope some of that helps.