Roll-forward #164317: Use bin/cache/engine.stamp (#164401)

... and this time, create `bin/cache` before trying to write to it!

(Tests updated to catch this regression)

.gitignore

@@ -1,9 +1,6 @@
 # Do not remove or rename entries in this file, only add new ones
 # See https://github.com/flutter/flutter/issues/128635 for more context.
 
-# This is dynamic in a monorepo
-bin/internal/engine.version
-
 # Miscellaneous
 *.class
 *.lock
@@ -33,6 +30,8 @@ bin/internal/engine.version
 /bin/cache/
 /bin/internal/bootstrap.bat
 /bin/internal/bootstrap.sh
+/bin/internal/engine.realm
+/bin/internal/engine.version
 /bin/mingit/
 /dev/benchmarks/mega_gallery/
 /dev/bots/.recipe_deps