I would say, that your two first statements are not wrong, i.e. you can defend them in a discussion if you explain what you mean by extension and special case.
Your conclusion, as you suspected anyway, is wrong however. Special relativity and Newtons law of gravitation are not the same thing. In fact they deal with different aspects of a physical theory.
Special relativity is better compared to Newtons laws of motion.
Both try to answer how objects move through time and space in the presence or
absence of external influences like forces.
Newton's law of gravitation is a way to get one of these forces that might act on bodies.
General relativity now combines both aspects into a single theory where a mass is moving 'forceless' in a geometry 'created' by all kinds of energy.
So if one says that general relativity is an extension of special relativity it possibly means that the equations of motion of a testbody look like the equations of special relativity in the right limit.
If you state that Newtons law of gravitation is a special case of general relativity, you are stretching the fact, that in the right limit the 'creation' part of general relativity looks like Newton's gravitational potential.
So to put it very short and colloquial, you are looking at different sites of an equation in these two statements.
Yes, special relativity is a special case of general relativity. General relativity reduces to special relativity, in the special case of a flat spacetime. I.e., general relativity reduces to special relativity, in the special case of gravity being negligible, for example in space far from any objects, or when considering a small enough piece of space in freefall that gravity is unimportant to the problem.
Like special relativity, general relativity also assumes that the speed of light is universal. However, when spacetime is curved, the universality of the speed of light can only be applied locally, within regions of spacetime that are small enough that the effects of gravity aren't important within the region.
Best Answer
Suppose we start by considering Galilean transformations, that is transformations between observers moving at different speeds where the speeds are well below the speed of light. Different observers will disagree about the speeds of objects, but there are some things they will agree on. Specifically, they will agree on the sizes of objects.
Suppose I have a metal rod that in my coordinate system has one end at the point $(0,0,0)$ and the other end at the point $(dx,dy,dz)$. The length of this rod can be calculated using Pythagoras' theorem:
$$ ds^2 = dx^2 + dy^2 + dz^2 \tag{1} $$
Now you may be moving relative to me, so we won't agree about the position and velocity of the rod, but we'll both agree on the length because, well, it's a chunk of metal - it doesn't change in size just because you are moving relative to me. So the length of the rod, $ds$, is an invariant i.e. it is something that all observers will agree on.
OK, let's move onto Special Relativity. What Special Relativity does is treat space and time together so the distance between two points has to take the time difference between the points into account as well. So our equation (1) is modified to include time and it becomes:
$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 \tag{2} $$
Note that our new equation for the length $ds$ now includes time, but the time has a minus sign. We also multiply the time by a constant with the dimensions of a velocity to convert the time into a length. Just as before the quantity $ds$ is an invariant i.e. all observers agree on it no matter how they are moving relative to each other. In fact we give this spacetime length a special name - we call it the proper length (or sometimes the proper time).
By now you're probably wondering what on Earth I'm rambling about, but it turns out we can derive all the weird stuff in Special Relativity simply from the requirement that $ds$ be an invariant. If you're interested I go through this in How do I derive the Lorentz contraction from the invariant interval?.
In fact the equation for $ds$ is so important in Special Relativity that it has its own name. It's called the Minkowski metric. And we can use this Minkowski metric to show that the speed of light must be the same for all observers. I do this in my answer to Special Relativity Second Postulate.
So where we've got to is that the fact the speed of light is constant in SR is equivalent to the statement that the Minkowski metric determines an invariant quantity. What General Relativity does is to generalise the Minkowski metric, equation (2). Suppose we rewrite equation (2) as:
$$ ds^2 = \sum_{\mu=0}^3 \sum_{\nu=0}^3 \,g_{\mu\nu}dx^\mu dx^\nu $$
where we are using the notation $dt=dx^0$, $dx=dx^1$, $dy=dx^2$ and $dz=dx^3$, and $g$ is the matrix:
$$g=\left(\begin{matrix} -c^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right)$$
This matrix $g$ is called the metric tensor. Specifically the matrix I've written above is the metric tensor for flat spacetime i.e. Minkowski spacetime.
In General Relativity this matrix can have different values for its entries, and indeed those elements can be functions of position rather than constants. For example the spacetime around a static uncharged black hole has a metric tensor called the Schwarzschild metric:
$$g=\left(\begin{matrix} -c^2(1-\frac{r_s}{r}) & 0 & 0 & 0 \\ 0 & \frac{1}{1-\frac{r_s}{r}} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2\sin^2\theta \end{matrix}\right)$$
(I mention this mostly for decoration - understanding how to work with the Schwarzschild metric needs you to do a course on GR)
In GR the metric $g$ is related to the distribution of matter and energy, and it is obtained by solving the Einstein equations (which is not a task for the faint hearted :-). The Minkowski metric is the solution we get when there is no matter or energy present${}^1$.
The point I'm getting at is that there is a simple sequence that takes use from everyday Newtonian mechanics to General Relativity. The first equation I wrote down, equation (1) i.e. Pythagoras' theorem, is also a metric - it's the metric for flat 3D space. Extending it to spacetime, equation (2), moves us on to Special Relativity, and extending equation (2) to a more general form for the metric tensor moves us on to general relativity. So Special Relativity is a subset of General Relativity, and Newtonian mechanics is a subset of Special Relativity.
To end let's return to that question of the speed of light. The speed of light is constant in SR so is it constant in GR? And the answer is, well, sort of. I go through this in some detail in GR. Einstein's 1911 Paper: On the Influence of Gravitation on the Propagation of Light but you may find this a bit hard going. So I'll simply say that in GR the speed of light is always locally constant. That is, if I measure the speed of light at my location I will always get the result $c$. And if you measure the speed of light at your location you'll also get the result $c$. But, if I measure the speed of light at your location, and vice versa, we will in general not get the result $c$.
${}^1$ actually there are lots of solutions when no matter or energy is present. These are the vacuum solutions. The Minkowski metric is the solution with the lowest ADM energy.