Mathematics of Waves, Interpretation of Sine wave as a function of $x$

The right way to think about these sorts of signs is usually to freeze the wave at time $t=0$ and see what happens.

In your convention the function reduces to $$ begin{align} y(x, 0) & = A sinleft(-2pifrac xlambdaright) &= -A sinleft(2pifrac xlambdaright), end{align} $$ whereas in their convention the leading term is $+$ not $-$.

Ultimately both represent the exact same family of equations, one will just have negative $A$ when the other has positive $A$ and vice versa. In fact, your preference is more common among electrical engineers, who often like to deal with complex numbers by defining that $j=sqrt{-1}$ and then the complex rotation whose real projection is a wave is usually written $e^{j(omega t - k x)}.$ I know physicists who resolve this terminology difference in an amusing way, they say that which square root of $-1$ you choose (there are two of them) is arbitrary, and physicists and engineers have chosen the opposite ones so that $i=-j$, hah.

Physicists are more likely to describe it as $kx-omega t$, though. I am personally very weird and I hate writing parentheses in the exponent so I define the default argument of 1 as $1 = e^{2pi i}$ and then write absurd things like $1^{x/lambda - f t}$, or sometimes in my notes I write it ə or so. If I ever got back into physics and wanted to publish, my PhD advisor would likely smack me upside the head and tell me to write it a normal way.