Your PC used to be connected to the internet via an ADSL modem. The modem went online when you activated AOL, and went offline when you deactivated AOL.
A broadband router works differently. When first power the router up, it connects to the internet. You can turn your PC(s) on and off as many times as you want, but the router will always stay connected to the internet.
If you only have 1 PC, it's all a bit academic. If you have several PCs (maybe some wired and some wireless) then the router stays connected because it never knows what PC is going to want to do what!
is there any way in which I can ONLY go on-line when I wish to?
You could power the router off, or there is usually a function on an admin screen somewhere to disconnect / recycle the router. Not sure why you'd want to.
Additionally, the Tx/Rx arrow indicators on the task bar show constantly red with "DSL modem No Device" showing - no upload/download speed.
I think that's old software (relating to your old ADSL modem) that you no longer need. The ADSL modem isn't connected/isn't switched on any more, and the indicators are reporting that. Your PC doesn't care that the ADSL model has gone, because it is talking to the router instead.
RC