The result may say the quadratic factor is statistically significant, but with a value is $10^{−14}$ it is physically insignificant. 

I am thinking your question is "Why is the quadratic coefficient so small and not equal to 1 from the original definition?"  To answer this look at your plot.  The x range (from 0 to 1) is small and in that range the function appears linear. (no noticeable curvature). The contribution quadratic (or higher orders) is very small in the range of -1 < x < 1. Since the square of small number is even smaller. ) $.1^2 = 0.01$.  
See the blue and green lines below.  Both lines are nearly equal.

[![enter image description here][1]][1]

Now for your second model, fitting a quadratic without the linear term.   
The least squares method is finding the best 2 coefficients to minimize the error and the result is the red curve.  Again there are 50 data points in a small x-range where the quadratic effect is insignificant and thus still a pretty good fit.  
In fact you just fit the intercept (without any independent variable) you will still end up with a decent fit.  

Hopefully this help cleared up your confusion.

    Var1 <- seq(1:51) 
    Var2 <- seq(0, 1, 0.02)
    Var3 <- Var2^2
    Test <- data.frame(cbind(Var1, Var2, Var3))
    plot(Test$Var2, Test$Var1)
    
    linear <- lm(Var1 ~ Var2, Test)
    summary(linear)
    abline(linear, col="blue")
    
    quad <- lm(Var1 ~ Var2 + Var3, Test)
    summary(quad)
    lines(x=Test$Var2, y=predict(quad, Test), col="green")
    
    quad_lite <- lm(Var1 ~ Var3, Test)
    summary(quad_lite)
    lines(x=Test$Var2, y=predict(quad_lite, Test), col="red")
    
    intercept <- lm(Var1 ~ 1, Test)
    summary(intercept)

  [1]: https://i.sstatic.net/A6L87.png