So, although this is not a default option within the embed code box on YouTube videos, this can be done quite simply by adding the following to the end of a YouTube embed URL:
So for example, the standard iFrame embed code you would get for a video like the one above would be:
<iframe width="560" height="349" src="http://www.youtube.com/embed/VIDEOID" frameborder="0" allowfullscreen></iframe>
In this case, you merely need to add "?modestbranding=1" to the end of the src= tag like so:
... src="http://www.youtube.com/embed/VIDEOID?modestbranding=1" ...
Issues, Tips & Work Arounds:
This only works if you use it as the first parameter after the video id. IE - it can not be added after any other parameters or it will break and the logo will reappear when the video starts. To illustrate what I mean, you can NOT do something like this:
... src="http://www.youtube.com/embed/VIDEOID?hd=1&modestbranding=1" ...
The modestbranding parameter needs to come first.
Additionally, this will not work with a few of the other embed player parameters like "showinfo" which can be used to set whether or not the title shows at the top of the video (0= no title). For whatever reason, if you continue to use "showinfo=0", the "modestbranding" parameter will not function as it's supposed to and the logo will reappear when you press play.
As a work around to this, I discovered a new parameter that they dont publicize => "title=". So, if you would like to remove the logo AND you want to remove the title from the player, you need to structure the embed src player string as follows, leaving "title=" blank:
... src="http://www.youtube.com/embed/VIDEOID?modestbranding=1&title=" ...
As just a fun little note on this - I found out that you can actually change that title to be whatever you want. See this example.