As someone who doesn't remember a lot about learning to ride a bici as a child - but who does remember a lot about learning to ride a moto as an adult - I think parking lot practice and working with an instructor are very smart.
Falling over IS a real concern, falling over in turns even more so. Turning a two-wheeled vehicle involves knowing when and where to shift your weight and when and how much to apply power (including being in the right gear), and while it's unconscious and natural to someone who's been riding since childhood, it's a skill that you had to learn then, even if you don't remember how many times you fell over when you were seven.
Maybe for us non-skiers, learning to ski at 50 would be a good analogy (the likelihood of falling is obvious, and the percentage of us who learned as young children is much smaller). Good for you, Catrin, hope you're having fun and learning fast!
Speed comes from what you put behind you. - Judi Ketteler