Edinburgh, London, Paris and/or Amsterdam?

Personally, while I love Paris, I would stay with your original 2 cities. Will you be planning any day trips from Edinburgh or London? There is so much to see.

I don't think the transportation costs and change of hotels is worth it if you're only going to spend 1-2 days. And squeezing both Paris and Amsterdam in to your 2 weeks would be crowded.

Have you thought of adding an extra stop in another UK city. York for a few days would be a perfect combination with London and Edinburgh.

Amsterdam, hands down. There is plenty to do there, especially in a short stay (3-4 would be my preference, rather than 1-2). Anne Frank House and the Van Gogh Museum are fantastic. There are lots of other notable museums. Evening canal tour. Lots of interesting day trip options.
Eurostar to/from London makes it easy to get to.
Check out https://sweetshotel.amsterdam to see if it's right for you. I would have loved to stay there, but I was traveling with someone I didn't want to share a bed with. We stayed at the Nadia, which is a budget hotel in a great location. I posted a review: https://community.ricksteves.com/travel-forum/netherlands-reviews/hotel-nadia-recommendation-with-lots-of-caveats
Like you, I had been before and wasn't sure about going back, but when I did, I realized how much I love Amsterdam. It's beautiful, quiet, walkable, just a delightful place to be. I'd love to go back for a longer stay.
I like Paris just fine, and with more time, I would absolutely recommend it. But your trip is very short, and Amsterdam just seems like a better fit if you feel like you can't fill 15 days between London and Edinburgh.

A smart way to gain a day on the way to Amsterdam is to take the overnight ferry.
If starting from Edinburgh DFDS to Ijmuiden from North Shields (Newcastle).
If starting from London use the Dutchflyer - train from Liverpool Street to Harwich Port then Stena ferry to the Hook of Holland.
From the Hook the official route nowadays is tram to Schiedam for train to AMS. But no reason why you can't go via Rotterdam.

Instead of just doing LARGE cities, consider enjoying some of the British countryside and it smaller cities and towns.

From Edinburgh, head north to St. Andrews. Also, you could head further north to Lock Ness and Inverness and/or head west to the western highlands.

From London, consider Stonehenge, Bath, the Cotswolds, Bath, Salisbury, Avebury, Winchester, Canterbury, Oxford, Cambridge. Also, north enjoy the Yorkshire Moors and York with its ancient walls and the National Railway Museum as well as the Minster.

Amsterdam does have some great museums, the Rijksmuseum, Van Gogh Museum and Anne Frank Museum. You probably need more time to do Paris.

Only one of Paris or Amsterdam-if necessary, flip a coin to decide which one.

If you haven't been to Edinburgh, put more energy and time into making that a special component of your trip that is a new, shared memory, although there is probably a lot of London that could be new to you, too. There are many opportunities for day trips or longer excursions from both cities.

So, I guess I'm with Jean on keeping to your original plan.
If you really want an additional city, I would probably choose Amsterdam. It sounds as if you don't have the negative vibes to overcome, more opportunities in A for experiences new to you, and it's less overwhelming than Paris. Save Paris for a longer trip.

Does 15 days mean 16 nights? Or are you counting your arrival and departure days?

To me here’s how I’d want to divide time… but I am also a detail traveler not a big picture traveler.

Edinburgh: 5 nights, more if you want to do day trips. This gives you 4 full days to see the city. I just spent 3 nights there as an intermediate stop over and my 2 days were very full even without seeing some of the main sites.

London: 7 nights (6 full days) , more if you want day trips. Bletchley Park if you have a WWII interest? Windsor Castle? Hampton Court Palace? Kew Gardens?

One day for train travel between the two. You may have a half a day on this day at either end. Or the lines could be down” or the previous train might hut something and you might be delayed.

That only leaves a couple of nights for another location. York is on the train line between Edinburgh and London and is excellent for a 3 night stay! Or more!

For myself, Paris needs 7 nights, Amsterdam 5. YMMV.

Since you said 15 days including arrival and departure I calculate you only have 13 and a half day to sight see and that does not allow for a half day to get to London and a half day to get to Paris. Also your arrival day I considered a half day due to potential jet lag. I would therefore just stay with Edinburgh and London or Edinburgh and fly to Amsterdam. Then you have some quality and quantity time to enjoy two places. We did a similar trip about a year ago into Edinburgh , train to London then train to Paris and home over three weeks which was just enough. Just something to consider. Just be sure to book multi city airline tickets. Good kuck.