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The difference between a partial derivative and a total derivative is that the partial derivative measures the change in a function when only one of its arguments varies, while the total derivative measures the change in a function when all of its arguments vary. The $\frac{\partial z}{\partial x}$ notation is somewhat misleading in this way, because you can't detach the top from the bottom and still have the $\partial$ symbol make sense—$\partial z$ only has a meaning when you know *which* part of the change in $z$ is being measured, the part obtained by varying $x$ or the part obtained by varying $y$.

So your error is in simplifying $\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}$ to $\frac{\partial z}{\partial t}$, even though the notation suggests that this should be valid. The product $\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}$ actually represents just the change in $z$ per unit change in $t$ that is attributed to variation in $x$. If you add that quantity to the change in $z$ per unit change in $t$ that is attributed to variation in $y$, you get the total change in $z$ per unit change in $t$, which is the total derivative.