What is the name of the variant of ICP where the correspondence of the points are known?
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.