A summary of answers with example output
This is using a local repository with five simple commits.
‣ git log --name-only
commit ed080bc88b7bf0c5125e093a26549f3755f7ae74 (HEAD -> master)
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:46:04 2019 -0700
    mv file4 to file5
file5
commit 5c4e8cfbe3554fe3d7d99b5ae4ba381fa1cdb328
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:36:32 2019 -0700
    foo file1
    really important to foo before the bar
file1
commit 1b6413400b5a6a96d062a7c13109e6325e081c85
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:34:37 2019 -0700
    foobar file2, rm file3
file2
file3
commit e0dd02ce23977c782987a206236da5ab784543cc
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:33:05 2019 -0700
    Add file4
file4
commit b58e85692f711d402bae4ca606d3d2262bb76cf1
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:32:41 2019 -0700
    Added files
file1
file2
file3
‣ git log --name-status
commit ed080bc88b7bf0c5125e093a26549f3755f7ae74 (HEAD -> master)
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:46:04 2019 -0700
    mv file4 to file5
R100    file4   file5
commit 5c4e8cfbe3554fe3d7d99b5ae4ba381fa1cdb328
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:36:32 2019 -0700
    foo file1
    really important to foo before the bar
M       file1
commit 1b6413400b5a6a96d062a7c13109e6325e081c85
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:34:37 2019 -0700
    foobar file2, rm file3
M       file2
D       file3
commit e0dd02ce23977c782987a206236da5ab784543cc
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:33:05 2019 -0700
    Add file4
A       file4
commit b58e85692f711d402bae4ca606d3d2262bb76cf1
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:32:41 2019 -0700
    Added files
A       file1
A       file2
A       file3
‣ git log --stat
commit ed080bc88b7bf0c5125e093a26549f3755f7ae74 (HEAD -> master)
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:46:04 2019 -0700
    mv file4 to file5
 file4 => file5 | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
commit 5c4e8cfbe3554fe3d7d99b5ae4ba381fa1cdb328
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:36:32 2019 -0700
    foo file1
    really important to foo before the bar
 file1 | 3 +++
 1 file changed, 3 insertions(+)
commit 1b6413400b5a6a96d062a7c13109e6325e081c85
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:34:37 2019 -0700
    foobar file2, rm file3
 file2 | 1 +
 file3 | 0
 2 files changed, 1 insertion(+)
commit e0dd02ce23977c782987a206236da5ab784543cc
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:33:05 2019 -0700
    Add file4
 file4 | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
commit b58e85692f711d402bae4ca606d3d2262bb76cf1
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:32:41 2019 -0700
    Added files
 file1 | 0
 file2 | 0
 file3 | 0
 3 files changed, 0 insertions(+), 0 deletions(-)
‣ git log --name-only --oneline
ed080bc (HEAD -> master) mv file4 to file5
file5
5c4e8cf foo file1
file1
1b64134 foobar file2, rm file3
file2
file3
e0dd02c Add file4
file4
b58e856 Added files
file1
file2
file3
‣ git log --pretty=oneline --graph --name-status
* ed080bc88b7bf0c5125e093a26549f3755f7ae74 (HEAD -> master) mv file4 to file5
| R100  file4   file5
* 5c4e8cfbe3554fe3d7d99b5ae4ba381fa1cdb328 foo file1
| M     file1
* 1b6413400b5a6a96d062a7c13109e6325e081c85 foobar file2, rm file3
| M     file2
| D     file3
* e0dd02ce23977c782987a206236da5ab784543cc Add file4
| A     file4
* b58e85692f711d402bae4ca606d3d2262bb76cf1 Added files
  A     file1
  A     file2
  A     file3
‣ git diff-tree HEAD
ed080bc88b7bf0c5125e093a26549f3755f7ae74
:100644 000000 e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 0000000000000000000000000000000000000000 D  file4
:000000 100644 0000000000000000000000000000000000000000 e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 A  file5
‣ git log --stat --pretty=short --graph
* commit ed080bc88b7bf0c5125e093a26549f3755f7ae74 (HEAD -> master)
| Author: My Name <user@email.com>
| 
|     mv file4 to file5
| 
|  file4 => file5 | 0
|  1 file changed, 0 insertions(+), 0 deletions(-)
| 
* commit 5c4e8cfbe3554fe3d7d99b5ae4ba381fa1cdb328
| Author: My Name <user@email.com>
| 
|     foo file1
| 
|  file1 | 3 +++
|  1 file changed, 3 insertions(+)
| 
* commit 1b6413400b5a6a96d062a7c13109e6325e081c85
| Author: My Name <user@email.com>
| 
|     foobar file2, rm file3
| 
|  file2 | 1 +
|  file3 | 0
|  2 files changed, 1 insertion(+)
| 
* commit e0dd02ce23977c782987a206236da5ab784543cc
| Author: My Name <user@email.com>
| 
|     Add file4
| 
|  file4 | 0
|  1 file changed, 0 insertions(+), 0 deletions(-)
| 
* commit b58e85692f711d402bae4ca606d3d2262bb76cf1
  Author: My Name <user@email.com>
      Added files
   file1 | 0
   file2 | 0
   file3 | 0
   3 files changed, 0 insertions(+), 0 deletions(-)
‣ git log --name-only --pretty=format:
file5
file1
file2
file3
file4
file1
file2
file3
‣ git log --name-status --pretty=format:
R100    file4   file5
M       file1
M       file2
D       file3
A       file4
A       file1
A       file2
A       file3
‣ git diff --stat 'HEAD^!'
 file4 => file5 | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
‣ git show
commit ed080bc88b7bf0c5125e093a26549f3755f7ae74 (HEAD -> master)
Author: My Name <user@email.com>
Date:   Mon Oct 21 15:46:04 2019 -0700
    mv file4 to file5
diff --git a/file4 b/file5
similarity index 100%
rename from file4
rename to file5
Credits to @CB-Bailey @Peter-Suwara @Gaurav @Omer-Dagan @xsor @Hazok @nrz @ptc