This has opened up a huge can of worms.
The first thing to note is that there is no difference between fares from Richmond Underground and Richmond (London) Rail stations. It’s all one station but TfL have this obsession with showing multiple instances of stations if they have more than one mode.
The next thing is that there is no out-of-station interchange in Windsor. So far none have been set up beyond zone 6, although issues with intermediate gatelines mean that we should see internal OSIs at Stevenage and Welwyn Garden City soon. This explains why your via Windsor journey was split into two – it was actually two journeys according to TfL.
West Drayton to Reading and branches were the last contactless extensions set up before project Oval, as I mentioned above, while Feltham to Windsor is very much part of Oval. However, given that there is no OSI in Windsor we can say that fares between Richmond and Slough are set up differently and there’s a reasonable chance that National Rail and TfL won’t show the same fare.
We can get some clues by looking at my fare finder (driven by TfL open data) because I add zonal coverage for journeys set up before Oval. The default route for Richmond to Slough is zones 3-16 which as suggested means District via Turnham Green to Ealing Broadway. The alternative routes involve taking SWR to South London terminals or Vauxhall or Queenstown Road Battersea and then taking various Underground lines to Paddington. Note that the change between Underground and National Rail at Paddington does not trigger the alternative route, which leads us onto …
About that can of worms. The National Rail journey planner thinks Richmond to Willesden Junction on Overground, then Overground/Bakerloo to Paddington with an optional change at Queens Park, then GWR to Slough is the second best option after Windsor. Because the OSI between Paddington LU and Paddington NR is not used in the fare setting, the fare charged will be the default fare. Indeed, it’s possible to do it without passing through a gateline because the Bakerloo line and Elizabeth line stations at Paddington are linked inside the paid area.
In summary, I’ve no idea where National Rail is getting the £9.70 fare from. You will be charged according to the TfL driven fare finders either £10.20 via South London terminals, £8.70 if you make two journeys via Windsor, or £4.50 via any other route (but don’t leave the PAYG area via Ascot and Reading).