My Level/Background:
I have just completed my first year of undergrad. In high school, I completed AP Physics C Mechanics and Electricity and Magnetism. In my first year of undergrad, I completed a course on Newtonian Mechanics and a course on Special Relativity and Electromagnetism which both approximately followed the sections on those topics in the Feynman Lectures on Physics.
The Question
I am starting to dive into tensor analysis and general relativity in my free time and I'm having some confusion about the Einstein Field Equation.
The Einstein Field Equation (without cosmological constant) states that $G_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}$ where $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}$ is the Einstein curvature tensor.
In most pop-science explanations of GR they say that matter and energy (or their density and flow I guess), which are represented by $T_{\mu\nu}$, cause spacetime to curve, which I assume is represented by the curvature tensor $G_{\mu\nu}$. Objects then move along the shortest path proper time path (geodesic) in this distorted spacetime.
They often do this by giving the pretty misleading picture of placing a large mass on a trampoline, where the fabric of the trampoline is spacetime, and showing how the large mass causes the fabric to bend and how this affects the motion of smaller objects thrown on the trampoline.
In the case of a spherical non-rotating planet, I'm assuming $T_{\mu\nu}$ is $0$ everywhere except for where the planet is. So that means $G_{\mu\nu} = 0$ everywhere not inside the planet.
My question is does that mean there is no curvature outside the planet (or is Einstein curvature a different thing than regular curvature)? Since this seems to imply that there would be no curvature in spacetime outside the planet which is clearly wrong since objects do orbit the Sun.
Or does the value of $T_{\mu\nu}$ inside the planet (where it is nonzero) affect the curvature of spacetime outside the planet (where it is zero) in a large radius around it?
In summary, what is the best way to think about how mass and energy affect the curvature of spacetime around them?