Catch yourself with your feet biking? I take it you do not bike much.
So right off, running up stairs you trip you do not get much hurt. Down you do need to be careful; I hold the handrails when possible.
My experience is two different sets of outdoor stairs that are on two of my preferred run routes that I often throw into the mix when I go past as interval training. Also the stairs from my basement that I run up and down carrying dumbbells as part of my work outs off and on. Have not fallen once. Once did miss a “box jump” onto a concrete barrier and scraped my shin up pretty bad - that was embarassing to be dripping blood on the run home. Biking I’ve been hit by cars and gone over my handlebars having to stop suddenly, albeit thanks to my helmet nothing worse than an AC separation and scrapes.
I don’t see it as mentioned yet, but you can do squats, deadlifts, leg presses non-stop until you die, but if you want to see definition you need to attend more to your diet. Since you said that is your desire, you need to focus more on losing fat? Coupled with beginning squat and DL training, it should go well.
I think that you should listen to the advice you’re paying for, from the professional who knows your situation far better than any of us.
Though if you insist on listening to us instead of him I understand if you want definition, you need more cardio (and better diet), not more quad-only strength. For a good leg burn and cardio combined, I suggest a half-dozen shuttle-runs at full speed.
There is no way to get a specific body part defined. Where you gain and lose fat first is genetically determined, and not alterable by exercise. Working your legs, no matter how you do it, does not reduce the fat levels in your legs.
Definition is a combination of muscle mass throughout the various motor units of a particular muscle and how much fat covers it up.
Our op volunteered in post#5 that he knows that a lower fat percentage is part of the equation (stated as a body weight target “which would be a relatively lean weight for me”) and others, from PunditLisa on, have reinforced the importance of fat loss to his achieving that “definition” goal. The question is what can he do in way of exercise that keeps his legs muscle mass increasing. Some here claim that squats with heavy weights are the only way to do that. I dispute that claim. The goal can be accomplished not only without heavy weight squats but even with body weight exercises alone.