Type of tour

I don't understand either. Why not have normal cycling? 

Each profile has different settings which are relevant to the names they are given.  "Indoor" has the GPS off.  "Road" will navigate via roads instead of off road trails. "Mountain" will have the mountain bike specific features turned on.

You are free to get rid of the profiles the unit comes with and you are free to create new profiles.  Just create a new profile called "Bike Ride" if that is what you want to choose.

Yep.  The labels on Garmin Connect are just labels.  The activity profiles on the Edge define how the edge is configured and it behaves for those activities.

You're free to create your own profiles, delete existing profiles, and copy/edit/rename any existing profiles.  The simplest thing to do is identify an existing profile that's closest to what you want, copy it, name it, and edit the copy (data screens, number and type of data fields, routing preferences, maps, automatic features, etc...) to be exactly what you want, then, if you want, delete the original.  All of the profiles on my Edge are ones I created/edited to be exactly what I want for the different types of riding I do.

No, I’m not talking about the activity profiles wich define an activity. I am talking about the label you can change when you finish the activity, press stop and then change under the summary screen before you hit save. 

I never even noticed you can do that.  I just checked it out on my 830.  I don't know if you can edit that list.

That ride classification list is fixed and defined in the FIT format for cycling sub-sport 

but if the Fenix/Forerunner can just name it Cycling like the main sport, why not the Edge?

maybe because Fenix and Forerunner are general porpose devices, but Edge is just for cycling and to name activity just cycling would say nothing.

The Edge does name it Cycling as the main sport, but then adds additional information via the sub-sport type.

You can change the classification in Garmin Connect if you wish, but you would need to do it for each activity.

In Garmin connect, any cycling category will be included under the general Cycling category.  When looking at the activities list, you can select Cycling and all cycling categories will be included in the list.  You can also then select the specific category to see only that category listed.  The same goes for doing summary reports. 

So, irrespective of what category of cycling it's identified as on the Edge, it will always show up under the general Cycling Category on GC. And, as already mentioned, you can change the category of an activity on GC and move it from one of the sub-categories of cycling to the general Cycling category.

That said, AFAIK, you can't have an activity on the Edge not be automatically categorized as one of the sub categories of Cycling, which seems to be what you're looking for.  However, since you don't care about sub categories, whichever sub category yours are identified as, they will still always be listed on the general Cycling category in GC.