[doc] Minor updates to Docker/Apt/S3 release documentation (#19787)

doc/_pages/release_playbook.md

@@ -186,7 +186,7 @@ the main body of the document:
    2. Click "Publish release"
    3. Notify `@BetsyMcPhail` by creating a GitHub issue asking her to manually 
       tag docker images and upload the releases to S3. Be sure to provide her 
-      with the binary date and release tag in the same ping.
+      with the release tag in the same ping.
    4. Announce on Drake Slack, ``#general``.
    5. Party on, Wayne.
 

tools/release_engineering/dev/README.md

@@ -75,6 +75,20 @@ use `v` on the version string. For example:
 
     bazel run //tools/release_engineering/dev:push_release -- 1.0.0
 
+### Verification
+
+Verify that:
+
+1. [s3://drake-packages/drake/release](https://s3.console.aws.amazon.com/s3/buckets/drake-packages?region=us-east-1&prefix=drake/release/&showversions=false)
+contains a set of `drake-<version>-[...].tar.gz[...]` files for each supported
+configuration (e.g. Focal, Jammy and Mac).
+
+1. [Dockerhub](https://hub.docker.com/r/robotlocomotion/drake/tags?ordering=last_updated&page=1)
+has `<version>` tags for each supported configuration (e.g. Focal and Jammy).
+
+1. The `*.deb` files are in AWS
+`S3/Buckets/drake-packages/drake/release/<configuration>/drake-dev_<version>-1_amd64.deb` for each supported configuration (e.g. focal and jammy)
+
 ## Run script for apt
 
 (Before proceeding, refer to the sections below if you need to add a new
@@ -88,25 +102,11 @@ Once your machine is set-up, run the `push_release` script as described below:
 The release creator will provide the version. Again, don’t use `v` on the
 version string. For example:
 
-    ./push_release 0.32.0 20210714 --apt
+    ./push_release 0.32.0
 
 The script may prompt for the GPG passphrase, which may be found in the AWS
 Secrets Manager. The script may prompt for this multiple times.
 
-### Verification
-
-Verify that:
-
-1. [s3://drake-packages/drake/release](https://s3.console.aws.amazon.com/s3/buckets/drake-packages?region=us-east-1&prefix=drake/release/&showversions=false)
-contains a set of `drake-<version>-[...].tar.gz[...]` files for each supported
-configuration (e.g. Focal, Jammy and Mac).
-
-1. [Dockerhub](https://hub.docker.com/r/robotlocomotion/drake/tags?ordering=last_updated&page=1)
-has `<version>` tags for each supported configuration (e.g. Focal and Jammy).
-
-1. The `*.deb` files are in AWS
-`S3/Buckets/drake-packages/drake/release/<configuration>/drake-dev_<version>-1_amd64.deb` for each supported configuration (e.g. focal and jammy)
-
 ### [Optional] Add a new configuration
 
 For example, to add Jammy: