The Fitbit Surge is the $250 GPS unit with HRM; the Charge HR is the $150 unit with HRM and no GPS. My girlfriend has had the Charge HR for a few months now. The HRM is accurate most of the day but not so much for more intense exercise like riding uphill.

Why did your doctor recommend a Fitbit to you?

A lot of people (doctors and others) have recommendations like park at the far end of the lot when you get groceries. It'll have you walk an extra 200 feet and you won't get any dings in your paint. But walking really isn't great exercise, it doesn't get your ticker into even Zone 1 and you don't burn many calories. It sounds like you're in good shape and don't have a weight problem, or much of one. Maybe the doc just recommends this to everybody?

That said you might consider not using an activity tracker on the bike, as you're already using Strava and don't want to double county your calories burned. Seems like the best use you'd make of a Fitbit type device would be the rest of the day when you're not on the bike.

If you have an Android phone, Google Fit will use the phone's accelerometer and supposedly sort out when you're walking, cycling, or running and credit you appropriately. I haven't used it and have no idea how well it works. But it's free.

The older Fitbit One goes in your pants pocket and picks up pedal rotations as strokes. The wrist mounted units shouldn't get many steps on the bike but it varies. My Mio Fuse registers no steps unless I hit some very rough pavement, my Garmin Fenix 3 shows a lot of steps from a bike ride.