Understanding the Round Metric and various other Metric Tensors on $S^2$.

Suppose we are dealing with a $2$-dimensional sphere.For simplicity we consider $S^2\subset \mathbb R^3$.Now we can give a parametrization on the sphere via the map $\varphi:(0,2\pi)\times(0,\pi)\to \mathbb R^3$,given by,

$\varphi(\theta,\phi)=(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)=(x(\theta,\phi),y(\theta,\phi),z(\theta,\phi))$

Now coordinate vector field induced by $\varphi$ on $S^2$ is given by $X_1(p)=D\varphi(p)(\frac{\partial}{\partial\theta}|_{\varphi(p)})$ and $X_2(p)=D\varphi(p)(\frac{\partial}{\partial \phi}|_{\varphi(p)})$ which turns out to be,

$X_\theta(\varphi(p))=\frac{\partial x}{\partial\theta}\frac{\partial}{\partial x}_{\varphi(p)}+\frac{\partial y}{\partial\theta}\frac{\partial}{\partial y}|_{\varphi(p)}+\frac{\partial z}{\partial\theta}\frac{\partial}{\partial z}|_{\varphi(p)}$

$X_\phi(\varphi(p))=\frac{\partial x}{\partial\phi}\frac{\partial}{\partial x}|_{\varphi(p)}+\frac{\partial y}{\partial\phi}\frac{\partial}{\partial y}|_{\varphi(p)}+\frac{\partial z}{\partial\phi}\frac{\partial}{\partial z}|_{\varphi(p)}$

Now if we consider the Euclidean metric on $\mathbb R^3$ then the inner products of these two coordinate vector fields (along $\varphi$) will be $g_{11}=\langle X_\theta,X_\theta\rangle,g_{12}=g_{21}=\langle X_\theta,X_\phi\rangle$ and $g_{22}=\langle X_\phi,X_\phi\rangle$ and calculating these inner products we get the matrix $$A=(g_{ij})_{2\times 2}=\begin{pmatrix} \sin^2(\phi)& 0\\0 & 1\\ \end{pmatrix}$$ and with this Riemannian metric tensor on $S^2$,as far as I have understood,we call $S^2$ as a Riemannian manifold with the Round Metric.

Notice that here we are using the inner product of $\mathbb R^3$ to calculate $g_{ij}$ and hence the round metric is the same as the induced metric on $S^2$ from Euclidean metric,is this correct?I was that just confused a little about these two metrics being the same.

Also I am curious to know what can be some other metric that I could have defined on $S^2$ which would be unusual compared to the round metric?I am ready to allow bizarre situations too,for once.