Glad you found something. I recommend you make sure to set the program for audio stream capture rather recording from the sound card.
The Replay Programs come with a program called Replay Converter which converts audio streams to different formats. Most station streams are mp3 anyway, so I don't use it that much. Replay Media Catcher can also capture on demand programs and record online video, as well as being a podcast client. They do have an editing program but it's not much and Audacity seems to have more in that department and is more comparable to Audition. Let me know if the free/ad-free version has any limitations to push you to pay.
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I've been looking at Audacity and the documentation. It looks like it only records from the sound card and does not do audio stream capture. It also looks like it does not have the capability for set recording start and stop times as you would with a DVR. With Replay you can pick a program to record out of one of the program guides, or enter the audio stream URL and recording start and stop times manually. When you set it up, you can also tell Replay to convert the stream to any format you want at the end of the recording period. They really have made it like a Tivo for the desktop and it appears much simpler to use.
IHeartRadio also includes recent Coast to Coast programs to stream on demand, so you don't even need to use any recording software (unless you plan to start and stop recording, use VSC or want to listen on a smartphone without running up your data usage).
PS: Replay Radio has a neat feature called URL Finder for when stations mask the web address of their audio stream or use Flash players. Just turn it on and then start streaming the station and the URL shows up in finder box.