Fix and update version checks (#3792)

Currently our version update checks aren't actually working; the script doesn't work correctly if no explicit --base-sha is passed, but that's always how CI is calling it.

Fixes flutter/flutter#79823 (and version checks in general)

This makes a number of changes:
- Fixes it to work without --base-sha
- Adds tests that it works in that mode
  - And tightens existing tests to require ToolExit, not just any error, to reduce false-positive test success
- Adds verbose logging of the checks being done, to make it easier to debug this kind of issue in the future
- Tightens the exception handling for missing previous versions to just the line that's expected to fail in that case
- Only allows missing versions when "publish_to: none" is set
  - Adds that everywhere it's missing
  - Standardize the format in the repo to "none" (instead of also having "'none'").
- Allows the use of NEXT in CHANGELOG as a way of gathering changes that are worth noting, but not
  doing a publish cycle for. (Replaces the plan of using -dev versions, since that's actually harder to implement,
  and more confusing.)
  - Ensures that we don't forget to clean up NEXT entries when bumping versions

url_launcher/example/pubspec.yaml

@@ -1,5 +1,6 @@
 name: url_launcher_example
 description: Demonstrates how to use the url_launcher plugin.
+publish_to: none
 
 dependencies:
   flutter:

url_launcher_linux/example/pubspec.yaml

@@ -1,5 +1,6 @@
 name: url_launcher_example
 description: Demonstrates how to use the url_launcher plugin.
+publish_to: none
 
 dependencies:
   flutter:

url_launcher_macos/example/pubspec.yaml

@@ -1,5 +1,6 @@
 name: url_launcher_example
 description: Demonstrates how to use the url_launcher plugin.
+publish_to: none
 
 dependencies:
   flutter:

url_launcher_windows/example/pubspec.yaml

@@ -1,5 +1,6 @@
 name: url_launcher_example
 description: Demonstrates the Windows implementation of the url_launcher plugin.
+publish_to: none
 
 dependencies:
   flutter: