Regardless of whether EU or US law and regulations apply, one thing I'd suggest is to not over think or over elaborate the information you provide to airport/airline personnel. There is a service you need, pre-boarding. That's all. It doesn't matter if you look like you need it or not. If you start explaining why you need it, or what aspects of it you need, you'll go nuts, and it won't help.
My simplified example: a disabled friend reported to me that she informed the airline that she could walk up a few steps, and she was distraught when it turned out that she was expected to walk up a full flight of steps. The problem here is that to the airline, it's binary: you can walk up steps, or you can't walk up steps. Nobody is going to keep track of the fact that you can walk up three steps but not up twenty. In this case the correct answer, when asked if you can walk up steps, is "No." Keep it simple. If accepting the wheelchair, despite the fact that you can walk, is going to get you the service you need, take the wheelchair.