Hi Schmoo - welcome to the forum and a Happy New Year to you!
I am pretty sure you have been caught out by what I call 'the rule of the three colours'. The colours on the powerbase, controller and of the driver/car in the app must all match. The default colours are red and green, so I always start with those and then add blue, yellow, orange & white in that order. Each controller should be connected to a colour on the powerbase & you also connect each car to a specific colour on the powerbase - these two processes are explained in the Quick Start Guide. Connect a car to red and you control it with the red controller - whether the app is on or off.
The third colour connection is to match the car ID (the colour on the powerbase the car is connected to) with the background colour of the driver tile in the race set-up menu of the app. This is the sliver of colour to the top and left of the tile. Set-up a car/driver on the red tile (default = Driver 1) and the powerbase lap sensors will search for the car connected to red on the powerbase and display data for it in the app. If you set up the car on another colour in the app, the red controller will still control the car connected to red, but no data will be recorded. It will seem as if the app isn't counting laps - however, it hasn't been set up to 'see' the car. It's the same for the calibration process. Remember, all three colours must match.
As for the apps - the 'Scalextric ARC' app is the current app - released in February/March 2018. It is designed to work with all three ARC powerbases (One, Air & Pro) and is being actively developed and de-bugged. The 'App Race Control (Legacy)' app is the original version and a completely different app - so both can be installed on the same smart device. The 'legacy' app was developed for ARC One and ARC Air before the release of ARC Pro in November 2017. The app is no longer being developed or debugged - it is a final version, available for those who prefer it or are using older devices that might struggle with the latest version of the app.
I hope that helps and you are up and running very soon.