Yes, the tutorials you are watching is ActionScript, it is a global all purpose animation language really. It is not exclusive to Adobe by any means whatsoever. FlashDevelop is just a different environment in which you put the code but without a timeline.
I'm currently working on learning AS3. I tried Flash and didn't really care for it. Right now I'm writing code in Kate (KDE Text Editor) and compiling it with mxmlc (Part of Adobe's Flex SDK.) This seems to be the best way for me. I'm going to do all of the graphical work in either GIMP, Karbon or Inkscape.
You can also get an ActionScript plugin for the Eclipse IDE. I haven't tried the plugin but I played around with Eclipse and C a few years back and it was alright.
You can get the FlexSDK here: http://opensource.adobe.com/wiki/display/flexsdk/Flex+SDK
Here's a pretty good tutorial on making a game using the Flex SDK: http://www.brighthub.com/hubfolio/matthew-casperson/blog/archive/2009/09/03/flash-game-development-with-flex-and-actionscript.aspx
You can get the ActionScript plugin for Eclipse here: http://sourceforge.net/projects/as3ide/
Omnihero, you might want to rethink having more than 1 AV program at a time. When you have multiple anti-virus programs at once they often conflict with each other and nullify any protection you'd get out of them. As well, the Flex SDK is a great choice for an old computer because mxmlc is a command line tool so it really doesn't take much to run. And you could write your code in Notepad or Wordpad if you wanted (any text editor would work, just save your classes as .as files.)
You asked earlier in the thread how someone learned to code. I'd suggest getting the Actionscript 3.0 Bible. You can get a cheap copy off Amazon used. You can also preview some of it here: http://books.google.com/books?id=SstX7mlHLckC&rintsec=frontcover#v=onepage&q&f=false
I've also found these sites to be useful resources for help with coding: http://www.how-to-code.com/ http://www.8bitrocket.com/ http://www.bit-101.com/ http://help.adobe.com/en_US/FlashPlatform/reference/actionscript/3/index.html
