V1 isn't just about an engine out situation. It's a decison speed. If I have ANY issue prior to V1, I'll have enough runway to stop the aircraft on the remaining runway. If an issue presents itself after passing V1, I no longer have the stopping distance required and if I decide to stop, I'm probably going to go off the end of the runway.
V1 is calculated prior to EVERY departure and includes many variables. It's factored for weight, weather, aircraft performance and config etc. The calculations are quite complicated behind the scenes. Luckily most of the hard work is done by the manufacturer and the Performance department. Aircraft acceleration is taken into account, that in turn depends on the weather, flap setting, thrust reduction (if any) etc. So they know what the airspeed should be at each point down the runway. At some point, we've used enough of the runway that a decision to stop will mean an overrun. That's all it is.
VR is the rotate speed. This is also a complicated calculation behind the scenes. Basically it is the safest speed to begin rotation. Yes, we could try to rotate earlier, but I'll have to explain why the tail hit the deck. On my fleet, V1 is usually about 5-15kts less than VR, but on a long runway it's often the same speed.
V2 is the speed to fly if we lose an engine once were airbourne.