The "Welcome" control will only appear if the user is logged
in anyway. If you configure and internal site and then an external with
anonymous access it will not appear, but a "sign in" option can appear
instead. You shouldn't have to worry about using code to hide the control.

The behavior you have described is what I have accomplished so far.

For the first phase of our project, we will be creating an internal
site that does not allow anonymous access. This site will be
extended as a public (internet zone) site that does allow anonymous
access. For the first phase of our project, we will not be providing
any content that requires authentication to the public. Having said
this, I would like to make it so that the "Sign In" link does not
even appear when the user is not authenticated (when accessing the
site via the public route).

Any suggestions?

this is how we hid the sign in link.
}
.ms-globallinks{
font-size:8pt;
background:transparent;
font-family:tahoma;
display:none;
}
.ms-globallinks,.ms-globallinks a{
color:#666666;
text-decoration:none;
font-family:tahoma;
display:none;
}

I have heard it been stated that it is recommended that master  pages
should not be modified using Sharepoint Designer?

My question is why? Does this apply to creating new master pages,
modifying existing master pages, or both?

The article  for creating a minimalist master page  that appears in the
Sharepoint Server 2007 SDK states to use Sharepoint Designer for
creating the master page. I don't understand why the SDK document
would tell me to do something that is not recommended.

I tried using Visual Studio 2005 to create the master page using the
steps in the article. There are references to controls in virtual
paths (such as ~/_controltemplates) that generate tons of error
messages. It seems like Sharepoint Designer is the more friendly
choice.

As a rule, modifying existing (Out of the Box) master  pages
is a bad thing whether you use SharePoint Designer or Visual Studio.
SharePoint Designer is the preferred option as you said it has more of a
SharePoint Context that Visual Studio. You can use Visual Studio if you so
desire but my personal preference is to use SharePoint Designer. In previous
beta versions of SharePoint designer whenever you edited Master Pages it
sometimes messed up the control references. This has seen ben fixed and
works like a treat!

My personal opinion is use SharePoint Designer to make any Master Pages
changes or when building your own and use Visual Studio when you really want
to get your hands dirty with some code.