Understanding SVM Kernels

Following Andrew Ng's machine learning course, he explains SVM kernels by manually selecting 3 landmarks and defining 3 gaussian function based on them. Then he says that we are actually defining 3 new features which are $f_1$, $f_2$ and $f_3$.

$\hskip0.9in$

And by applying these 3 gaussian functions on every input data: $$x=(x_1,x_2)\to \hat{x}=(f_1(x), f_2(x), f_3(x))$$

it seems that we are mapping our data from $\mathbb R^2$ space to a $\mathbb R^3$ space. Now our goal is to find a hyperplane in the 3 dimensional space, where our transformed data is linearly separable. Is my understanding correct? If not, how these 3 new features should be interpreted?

$\hskip1in$

In some blog posts, i have read that by using a gaussian kernel, we are mapping our data to an infinite dimensional space (where gaussian kernel computes the dot product of transformed input data) which contradicts with my above understandings.