The confusion here comes from trying to work in a range of $t$ or $\lambda$ values where there is no constraint on the regression.

In your example, at the perfect fit of the regression line the sum of the squares of the regression coefficients is 1. So the value of $t=2$ (or any value of $t$ that is 1 or greater) places no constraint on the regression. In the space of $\lambda$ values, the entire unconstrained regression is represented by $\lambda = 0$. There is no one-to-one correspondence between $t$ and $\lambda$ in the unconstrained regression; all values of $t$ of 1 or greater in this case correspond to $\lambda=0$. That was the region that you have been investigating.

Only a value of $t$ less than 1 will place a constraint on the regression, corresponding to positive values of $\lambda$. As the accepted answer to this page shows, the one-to-one correspondence between $t$ and $\lambda$ holds "when the constraint is binding," in your example for values of $t$ less than 1.

The confusion here comes from trying to work in a range of $t$ or $\lambda$ values where there is no constraint on the regression.

In your example, at the perfect fit of the regression line the sum of the squares of the regression coefficients is 1. So the value of $t=2$ (or any value of $t$ that is 1 or greater) places no constraint on the regression. In the space of $\lambda$ values, the entire unconstrained regression is represented by $\lambda = 0$. There is no one-to-one correspondence between $t$ and $\lambda$ in the unconstrained regression; all values of $t$ of 1 or greater in this case correspond to $\lambda=0$. That was the region that you have been investigating.

Only a value of $t$ less than 1 will place a constraint on the regression, corresponding to positive values of $\lambda$. As the accepted answer to this page shows, the one-to-one correspondence between $t$ and $\lambda$ holds "when the constraint is binding," in your example for values of $t$ less than 1.