Submitting a Zoom meeting recording in a Canvas assignment or Discussion is not difficult, but it's helpful to think of the process as consisting of two parts:

Recording your meeting.
Embedding the video into the assignment itself for submission.

Recording Your Meeting

Start your Zoom meeting and press the record button located at the bottom of the meeting window.
Be sure to select the option to "Record to the Cloud", rather than "Record on this Computer".
This is an important step and will save you a great deal of time!
You can press the stop button to end the recording, or, you can simply end the meeting. Either way the recording will have automatically uploaded to the Zoom Cloud.
When cloud processing is complete, you will receive an email notification from Zoom that your recording is available for use.

Note: Wait at least 20 minutes after receiving the notification before proceeding to the next step.

Embedding Your Recording in the Canvas Assignment or Discussion

Click the "Start Assignment" button in the top right-hand corner of the page.

(Note that for a Canvas Discussion you'll click "Reply", rather than "Submit Assignment" to begin the process.)
In the rich content editor that appears on the page, click the "App" icon to reveal the Course Video ( ) selection tool.
Click on "Course Video"
In the "Course Video" window that opens, click on the "Choose" tab.
Click on the folder selection field shown below and type in "Meeting Recordings".
Locate and select your recording from the list that appears.
Now, click the "Insert" button found at the bottom-right corner of the page.
The Course Video window will close and your meeting video will appear inside of the Rich Content Editor of the assignment.
Finally, click the "Submit Assignment" button and you're done!