Screen capture software can record anything from your screen. I haven't seen a decent free screen capture software so far. Almost all of them are free to try.

You may need to read our hint on Avoiding Black screen when recording video with Screen Capture software from QuickTime player.

Also I suggest you to learn to get stream URLs using URL Snooper & Packet Sniffer software. Free Project URL Snooper is the URL snooper software I recommend.

Unlike RTSP Quick Time (QT) streams, QT HTTP streams can be downloaded by any download manager. However HTTP protocol allows software to send a so called User Agent within an HTTP request. So web-site owners can identify you by user agent.
Many download manager allow you to change the user agent. But first you need to learn what kind of user agent your player or embedded player uses.
You may do this by using a packet sniffer & URL snooper software.

p.s. It is better to experiment with recording at the end of your membership.
There is no way to identify that you record with screen capture software.