Your guess is correct, If you created the variable inside __construct (or any other function) it would only be available inside that, not available for the whole class.
The example you are working on isn't a really good one to demonstrate this point. Where it is really noticeable is when you have more than one method (function) that uses the variable.
As a personal recommendation, for learning OOP with PHP, this is an awesome book:
http://www.amazon.com/Objects-Patterns- ... 1430260319 I know lots of people like "free" resources, but this is one of the few where I say the $25 for the electronic version is well worth it. (myself, I use books24x7.com and read it there, if you are in school somewhere, see if their library services have access to it or something similar)