What is the name of the variant of ICP where the correspondence of the points are known? Traditional ICP assumes that the inputs are point clouds with no predefined correspondences between the clouds.

If I know before hand that some / all points in one point cloud corresponds to some / all points in the other point cloud, I assume this becomes an easier problem to solve.

What is the name of this modified version of point set registration and what are some algorithms that solves this?
This problem is called "rigid registration", the correspondences being the "rigid" part.

As D.W. pointed out, you can do a least squares fit (or use something like RANSAC to find the correspondences and then do a least squares fit). In the absence of noise, the least squares fit has a particularly nice formulation which has been known for a long time.

If $\mathbf{x}_i$ and $\mathbf{y}_i$ are the two corresponding sets of coordinates (there are $n$ points), we want to find the rotation matrix $R$ and translation vector $\mathbf{t}$ which minimises:

$$\hbox{arg min} \frac{1}{n}\sum_i \left\Vert \mathbf{y}_i - R \mathbf{x}_i - \mathbf{t}\right\Vert^2 $$

Calculate the covariance matrix:

$$\mathbf{\mu}_x = \frac{1}{n} \sum_i \mathbf{x}_i$$
$$\mathbf{\mu}_y = \frac{1}{n} \sum_i \mathbf{y}_i$$
$$\Sigma_{xy} = \frac{1}{n} \sum_i (\mathbf{y}_i - \mathbf{\mu}_y) (\mathbf{x}_i - \mathbf{\mu}_x)^\intercal$$

Compute the singular value decomposition of the covariance matrix:

$$\Sigma_{xy} = U S V^\intercal$$

Then:

$$R = V U^\intercal$$

This algorithm can be made more robust in the presence of noise. There are also more recent algorithms which are competitive.