The Cambridge Dictionary, as quoted, clearly shows the differences and the similarities between these two words.
To summarise the meaning of resumption is limited to
the start of something again after it has stopped
Whilst Renewal relates to:
renewal noun (CONTINUE); the act of making an official document, agreement, or rule continue for more time:
renewal noun (START AGAIN); the act of starting again or starting to do something again:
renewal noun (IMPROVEMENT); a process in which something improves or is improved after being in a bad condition :
Only the second meaning is similar to that of resumption. Even this use is not exactly the same. Start again can equally relate to starting over (new approach, design, etc) as it can to resuming a paused process. However, "resumption" has been defined as a re-start not "starting over".
I would also add that I believe that most people using renewal meaning "start again", would actually be using it to relate to starting over or as it would be termed in the UK "starting afresh".
start over C.E.D.— phrasal verb with start verb US
(UK start afresh)
to begin to do something again, sometimes in a different way: