Drupal Site Upgrades with Drush in Git/Eclipse Workflow

I've got a Drupal 6.14 installation recently initialized as a Git repo that I now need to upgrade (from Drupal core 6.14 to 6.19 and to the latest versions of several contributed modules.  To safely upgrade this site I'll perform the upgrades locally and make sure they don't break any of the functionality - especially in the features from custom modules which may have been written against these now outdated versions of some important modules.  So the idea is to create a branch from the current code to a new line of development for these upgrades.  I'm familiar with SVN branches, but have been told I need to think differently about branches in Git.  So here it goes.

Master branch is (by default?) set as the reference for HEAD.  I think I'll create a branch of it called dev and perform upgrades there.  I've setup my Git repo in Eclipse using the EGet plugin.

I've got one local branch master (marked as currently checked out by the checkmark icon) which references the remote branch origin/master.

Right-clicking master, I select Create a new local branch...

I name it dev and leave the Checkout new branch option selected.  This makes any local changes apply to the newly created branch.  Makes sense.

So now my dev branch is marked as active/checked out.  The icon next to the local master branch indicates it has been merged (?).

If I wanted to switch to the master branch, I could right-click and select the Check out option.

Now that I've got a new branch to work on, I jump into Terminal and run drush update command on the site.

While this shows me the status of all enabled modules (many of them requiring SECURITY UPDATES), it only updates drupal core.  I really do want to continue (y).