Generally, you would want to do one of:

a) Keep any code that you've pulled in from another project in its own directory structure with a license file indicating where it came from and its licensing terms.

b) If you intend to modify the code or integrate it more tightly with your own, copy the notice into each source file that was taken and perhaps put a pre-amble along the lines of "Portions of this file were copied from XXX under the MIT license as follows:". Ideally you would make a commit with the file in its initial state as copied, and then if you ever need to determine what came from where and how it was licensed, it shouldn't be too difficult.

From my own past experience working with F/OSS at Microsoft, they should at the minimum have "third party notices" file somewhere in the repo. Something like this: https://github.com/microsoft/debugpy/blob/main/src/debugpy/T...

What you're proposing, updating the license file to list the authors, is a pretty common way to do this. It does mean that the code is mixed a bit, so it would be hard to split who owns what, but this is only relevant if one of the copyright owners wants to change the license (as they can legally only do that to the code they own).