It is not "this causes that" - all is happening together. Let me explain:
For me to understand aerodynamics, it helped to disregard all that talk of vortices and induction, but focus on the pressure field around a wing. When the theory of flight was developed, electricity was new and exciting, and it just happened that electric induction could be transferred to lift. Now every author still copies the explanations from a century ago, but they are totally unintuitive.
Every air molecule is in a dynamic equilibrium between inertial, pressure and viscous effects. Inertial means that the mass of the particle wants to travel on as before and needs force to be convinced otherwise. Pressure means that air particles oscillate all the time and bounce into other air particles. The more bouncing, the more force they experience. Viscosity means that air molecules, because of this oscillation, tend to assume the speed and direction of their neighbors.
Now to the airflow: When a wing approaches at subsonic speed, the low pressure area over its upper surface will suck in air ahead of it. See it this way: Above and downstream of a packet of air we have less bouncing of molecules (= less pressure), and now the undiminished bouncing of the air below and upstream of that packet will push its air molecules upwards and towards that wing. The packet of air will rise and accelerate towards the wing and be sucked into that low pressure area. Once there, it will "see" that the wing below it curves away from its path of travel, and if that path would remain unchanged, a vacuum between the wing and our packet of air would form. Reluctantly, the packet will change course and follow the wing's contour, but not without spreading out (= pressure loss). Spreading happens in flow direction - the packet is distorted and stretched lengthwise, but contracts in the direction orthogonally to the flow. This fast-flowing, low-pressure air will in turn suck in new air ahead and below of it, will go on to decelerate and regain its old pressure over the rear half of the wing, and will flow off with its new flow direction.
A packet of air which ends up below the wing will experience less uplift and acceleration, and in the convex part of highly cambered airfoils it will experience a compression. It also has to change its flow path, because the cambered and/or inclined wing will push the air below it downwards, creating more pressure and more bouncing from above for our packet below the wing. When both packets arrive at the trailing edge, they will have picked up some downward speed.
Behind the wing, both packets will continue along their downward path for a while due to inertia and push other air below them down and sideways. Above them, this air, having been pushed sideways before, will now fill the space above our two packets. Macroscopically, this looks like two big vortices. But the air in these vortices cannot act on the wing anymore, so it will not affect drag or lift. See here for more on that effect, including pretty pictures.
What is lift?
Following the picture of a pressure field outlined above, lift is the difference of pressure between upper and lower surface of the wing. The molecules will bounce against the wing skin more at the lower side than at the upper side, and the difference is lift.
Or you look at the macroscopic picture: A certain mass of air has been accelerated downwards by the wing, and this required a force to act on that air. This force is what keeps the aircraft up in the air: Lift.
Either way, you will arrive at the same result. By the way: Most of the directional change happens in the forward part of the airfoil, not at the trailing edge!
The misconception about those "wingtip vortices" and induced drag is hard to eradicate. Most authors copy what has been written before without clearly understanding the issue. Therefore I repeat it here again: Induced drag is the backward-pointing part of the pressure force vector. The vortices are only a consequence of downwash, which in turn is a consequence of lift creation. At the same speed, more induced drag is indeed linked to more lift, but the causality is different: Lift and induced drag are both part of the pressures acting on the wing. If you add up all the pressure forces acting on a wing, their resulting vector will point slightly backwards. The streamwise component is drag, and the component orthogonal to the direction of movement is lift. This is just a defininion, made for simplicity.