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I think that a family of counterexamples can be constructed in the following way. Note that my notation is slightly different from yours.

Start with a smooth, minimal projective surface $X$, and let $\tilde{X}$ be the blow-up of $X$ at a point $p \in X$. If $E$ is the exceptional divisor, $H$ is an ample divisor on $\tilde{X}$ and $n \in \mathbb{N}$ is big enough, a general divisor $D \in |nH| $ is smooth and irreducible and $\tilde{D}=D + E$ is ample, too.

Then $\tilde{X}-\tilde{D}$ is an affine variety,isomorphic to an affine subvariety $X^{0} \subset X$. Moreover, composing the isomorphism $X^{0} \to \tilde{X}-\tilde{D}$ with the inclusion $\tilde{X}-\tilde{D} \to \tilde{X}$, we get a birational morphism $\phi \colon X^{0} \to \tilde{X}$.

Such a morphism cannot be extended to a birational morphism $f \colon X \to \tilde{X}$. In fact, by minimality of $X$ such an extension would be an isomorphism, a contradiction since $X$ and $\tilde{X}$ have different Picard numbers.

I think that a family of counterexamples can be constructed in the following way. Note that my notation is slightly different from yours.

Start with a smooth, minimal projective surface $X$, and let $\tilde{X}$ be the blow-up of $X$ at a point $p \in X$. If $E$ is the exceptional divisor, $H$ is an ample divisor on $\tilde{X}$ and $n \in \mathbb{N}$ is big enough, a general divisor $D \in |nH| $ is smooth and irreducible and $\tilde{D}=D + E$ is ample, too.

Then $\tilde{X}-\tilde{D}$ is an affine variety, isomorphic to an affine subvariety $X^{0} \subset X$. Moreover, composing the isomorphism $X^{0} \to \tilde{X}-\tilde{D}$ with the inclusion $\tilde{X}-\tilde{D} \to \tilde{X}$, we get a birational morphism $\phi \colon X^{0} \to \tilde{X}$.

Such a morphism cannot be extended to a birational morphism $f \colon X \to \tilde{X}$. In fact, by minimality of $X$ such an extension would be an isomorphism, a contradiction since $X$ and $\tilde{X}$ have different Picard numbers.

I think that a family of counterexamples can be constructed in the following way.

Start with a smooth minimal surface $X$, and let $\tilde{X}$ be the blow-up of $X$ at a point $p \in X$. If $E$ is the exceptional divisor, $H$ is a very ample divisor on $\tilde{X}$ and $n \in \mathbb{N}$ is big enough, a general divisor $D \in |nH| $ is smooth and irreducible and $\tilde{D}=D + E$ is ample.

Then $\tilde{X}-\tilde{D}$ is an affine variety,isomorphic to an affine subvariety $X^{0} \subset X$. Moreover, composing the isomorphism $X^{0} \to \tilde{X}-\tilde{D}$ with the inclusion $\tilde{X}-\tilde{D} \to \tilde{X}$, we get a birational morphism $\phi \colon X^{0} \to \tilde{X}$.

Such a morphism cannot be extended to a birational morphism $f \colon X \to \tilde{X}$. In fact, by minimality of $X$ such an extension would be an isomorphism, a contradiction since $X$ and $\tilde{X}$ have different Picard numbers.

I think that a family of counterexamples can be constructed in the following way. Note that my notation is slightly different from yours.

Start with a smooth, minimal projective surface $X$, and let $\tilde{X}$ be the blow-up of $X$ at a point $p \in X$. If $E$ is the exceptional divisor, $H$ is an ample divisor on $\tilde{X}$ and $n \in \mathbb{N}$ is big enough, a general divisor $D \in |nH| $ is smooth and irreducible and $\tilde{D}=D + E$ is ample, too.

Then $\tilde{X}-\tilde{D}$ is an affine variety,isomorphic to an affine subvariety $X^{0} \subset X$. Moreover, composing the isomorphism $X^{0} \to \tilde{X}-\tilde{D}$ with the inclusion $\tilde{X}-\tilde{D} \to \tilde{X}$, we get a birational morphism $\phi \colon X^{0} \to \tilde{X}$.

Such a morphism cannot be extended to a birational morphism $f \colon X \to \tilde{X}$. In fact, by minimality of $X$ such an extension would be an isomorphism, a contradiction since $X$ and $\tilde{X}$ have different Picard numbers.

I think that a family of counterexamples can be constructed in the following way.

Start with a smooth minimal surface $X$, and let $\tilde{X}$ be the blow-up of $X$ at a point $p \in X$. If $E$ is the exceptional divisor and $H$ is a very ample divisor on $\tilde{X}$, for sufficiently big $n \in \mathbb{N}$ a general divisor $D \in |nH| $ is smooth and irreducible and $\tilde{D}=D + H$ is ample.

Then the affine varieties $X-D$ and $\tilde{X}-\tilde{D}$ are isomorphic, and composing the isomorphism with the inclusion $\tilde{X}-\tilde{D} \to \tilde{X}$ we get a birational morphism $\phi \colon X-D \to \tilde{X}$.

Such a morphism cannot be extended to a birational morphism $f \colon X \to \tilde{X}$. In fact, by minimality of $X$ such an extension would be an isomorphism, a contradiction since $X$ and $\tilde{X}$ have different Picard numbers.

I think that a family of counterexamples can be constructed in the following way.

Start with a smooth minimal surface $X$, and let $\tilde{X}$ be the blow-up of $X$ at a point $p \in X$. If $E$ is the exceptional divisor, $H$ is a very ample divisor on $\tilde{X}$ and $n \in \mathbb{N}$ is big enough, a general divisor $D \in |nH| $ is smooth and irreducible and $\tilde{D}=D + E$ is ample.

Then $\tilde{X}-\tilde{D}$ is an affine variety,isomorphic to an affine subvariety $X^{0} \subset X$. Moreover, composing the isomorphism $X^{0} \to \tilde{X}-\tilde{D}$ with the inclusion $\tilde{X}-\tilde{D} \to \tilde{X}$, we get a birational morphism $\phi \colon X^{0} \to \tilde{X}$.

Such a morphism cannot be extended to a birational morphism $f \colon X \to \tilde{X}$. In fact, by minimality of $X$ such an extension would be an isomorphism, a contradiction since $X$ and $\tilde{X}$ have different Picard numbers.