The name is wrong
Now the misconception, and it is close to universal: V1 is widely called the decision speed, as though it were the moment at which you make up your mind.
It is not. The distances above were computed on the assumption that by the time the airplane reaches V1, the rejection is already happening — the thrust coming back, the brakes going on. Everything after that instant is stopping distance. The recognition, the judgement, the decision and the first action all belong to the time before V1.
Which makes V1 something more uncomfortable than a decision point. It is the last speed from which stopping still fits on the runway you have. Arrive there still thinking, and the arithmetic that justified the number has already been invalidated.
The sensitivity is worth a moment, because the numbers are less forgiving than they look. On a takeoff where the runway is the limiting factor, rejecting from a V1 three knots higher than the correct one turns a stop that should have finished at the far end into one still doing about fifty knots when it arrives there, several hundred feet of runway too late. Three knots the other way is not free either: continue from too low a speed and the airplane crosses the far end at something nearer twenty feet than the thirty-five it was supposed to have. Neither error announces itself. Both are three knots.
Two distances are computed from every brake release — one to reject and stop, one to continue on a failed engine and get airborne — and because raising the failure speed lengthens the first while shortening the second, there is a speed where they meet, which is V1 and the shortest runway that works; balanced means only that those two came out equal on the day's conditions, so a wet runway moves the crossing rather than displacing V1 from it, while the balanced answer is also the shortest field that works, so every knot you deliberately move away from the crossing spends runway you must already have, buying weight, or a gentler rejection, or an option to reject that stays open longer; and it is not a moment to decide but the last speed from which stopping still fits, because the brakes must swallow the whole energy of a heavy airplane at speed, which is itself sometimes the limit on how heavy you may be.
How it shifts
Heavier: both distances grow, and the brakes are asked for more — so at some weight the rejection itself becomes the limiting case rather than the runway.
Wet: the stopping distance grows and the altitude wanted over the far end is lowered, so both strokes move and the crossing itself slides to a lower speed. The wet number is a lower balanced V1, not a displaced one.
Contaminated: worse in both directions at once — the braking is poorer still, and ploughing through the contaminant slows the acceleration as well, so V1 falls further and the weight usually comes off with it.
Downhill, or a tailwind: the going case is helped and the stopping case is hurt, so V1 comes down. Uphill or into wind, the reverse.
A stopway beyond the runway: only the stopping case benefits, so a higher V1 will carry more weight off that runway. A clearway: only the going case benefits, so a lower V1 does the same. Either way the gain is payload, not margin.
Hot or high: the same indicated speed is a greater true speed, so there is more energy to accelerate and more to dissipate, and both distances stretch.
More runway than the balanced field needs: the surplus is the only thing that can be spent, so V1 becomes a band rather than a number, and where you place it inside that band trades stopping margin against how long the option to reject stays open — as far, on a long enough runway, as the rotation speed itself. With no surplus there is no band.
If V1 is the decision speed, why is it too late to decide at V1?
Because it was never a decision speed, whatever it gets called. The stopping distance was computed on the assumption that the rejection is already under way as the airplane passes V1 — thrust closing, brakes applied. Everything before that instant is where the recognising and the deciding have to fit. Treat V1 as the moment to begin thinking and the number no longer describes your airplane, because the arithmetic that produced it assumed you had already acted.