Those descriptions aren't quite right. You can find the official descriptions in the documentation for the FileMetadata object, e.g., as returned by /2/files/upload.
In short, they're both the modified time of the file. The 'client_modified' time is the time as supplied by the client that uploaded the file, if any, and the 'server_modified' time is the time on the Dropbox servers when the file was uploaded. The Dropbox API does not return file creation time.
By the way, those are links to the documentation for the HTTPS endpoints themselves, but we recommend using one of the official SDKs if possible. Those have corresponding native methods for the HTTPS endpoints.
We love to learn from the educators who use Dropbox. Whether you teach kids, teens, adults or a combination of all three, we want to know what apps and integrations you use with Dropbox to help with teaching. Which of the ones below is your favorite, or most used tool?