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Edit: this counterexample does not satisfy the last condition.

Consider the subalgebra of $C^\infty(\mathbb{R})$ generated by $x^2$, then $I_0[A]$ consists of polynomials in $x^2$ with no constant term, and elements in $I_0^2[A]$ are polynomials in $x^2$ with lowest power $x^4$. So unless I misunderstood the question both hypothesis are false.

Edit: modified to satisfy the last condition.

Consider the subalgebra of $C^\infty(\mathbb{R})$ generated by two elements $x^2,xe^x$. This algebra separates points and "sees" any nonzero tangent vector. The ideal $I_0[A]$ has the functions $x^{2k+l}e^{lx}$ where $k,l\in\mathbb{N}$ and not both $k$ and $l$ are zero as a vector space basis and $I^2_0[A]$ is hence spanned by products of such elements. Using this one verifies that the function $x^2$ is not in $I^2_0[A]$. On the other hand it is contained in the rhs of both of the hypothesis.

Edit: modified to satisfy the last condition.

Consider the subalgebra of $C^\infty(\mathbb{R})$ generated by the two functions $x^2,e^x$. It separates points and notices each nonzero tangent vector of the line. Now for $a=0$ the function $x^2$ is included in the right hand sides of both of the hypothesis, but $x^2$ is not in $I^2_0[A]$.

Edit: this counterexample does not satisfy the last condition.

Consider the subalgebra of $C^\infty(\mathbb{R})$ generated by $x^2$, then $I_0[A]$ consists of polynomials in $x^2$ with no constant term, and elements in $I_0^2[A]$ are polynomials in $x^2$ with lowest power $x^4$. So unless I misunderstood the question both hypothesis are false.

Edit: this counterexample does not satisfy the last condition.

Consider the subalgebra of $C^\infty(\mathbb{R})$ generated by $x^2$, then $I_0[A]$ consists of polynomials in $x^2$ with no constant term, and elements in $I_0^2[A]$ are polynomials in $x^2$ with lowest power $x^4$. So unless I misunderstood the question both hypothesis are false.

Edit: modified to satisfy the last condition.

Consider the subalgebra of $C^\infty(\mathbb{R})$ generated by the two functions $x^2,e^x$. It separates points and notices each nonzero tangent vector of the line. Now for $a=0$ the function $x^2$ is included in the right hand sides of both of the hypothesis, but $x^2$ is not in $I^2_0[A]$.