You are right about the dropped $\sim$, it's probably just a typo. Furthermore, remember that in stochastic calculus, you have to take into account second order derivatives, i.e.
$$d\left(\frac{1}{Y_t}\right) = -\frac{1}{Y_t^2}dY_t + \frac{1}{2}\frac{2}{Y_t^3}dY_t^2$$
which is the Taylor expansion up to second order. Then you substitute $dY_t$ in the right hand side and take into account that
$$dY_t^2 = \gamma^2 Y_t^2 d\tilde{W}_t^2 = \gamma^2 Y_t^2 dt \; . $$
The reason you keep second order terms is because they might contain terms with quadratic variation proportional to $dt$. This is the case of Brownian motion itself which has $dW_t^2=dt$.
Extra on Taylor expansion:
Provided a function is sufficiently differentiable in some point of its domain, it is possible to approximate it by a polynomial in some neighborhood of that point. Say $f(x)$ around the point $a$ is approximately equal to
$$f(x) \approx f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2$$
Or if we put $x-a=dx$ and $f(x)-f(a)=df$ we can write this as
$$df \approx f'(a)dx+\frac{1}{2}f''(a)dx^2$$
This is true if the function is twice differentiable and some additional conditions which are a bit too technical to expand upon here.
