First one looks like the model is a DVD writer (2nd one looks like cd-rom), which means it should write DVD-R without issue. If Device Manager shows "no problem", then it could be a defective drive OR corrupted files associated with running the device. First, I would test it out, with a testing software using a new, blank DVD-R media inside. Try ImgBurn http://www.imgburn.com/index.php?act=download. Download it, and select Discovery mode to test writing capabilities of the optical drive. You want to isolate the issue to if the drive fails during writing OR if the device is treated as a writer at all.
Also, to create HP recovery media without a working, physical optical drive, i recommend a "virtual" optical drive software. I use Daemon Tools Lite, which is free for home use http://www.daemon-tools.cc/eng/downloads. That way, you can save an image of the recovery disc (.iso file), and burn it into a real cd or dvd using Imgburn at a later time. Very easy to do, and if never used before, you can google it and get information easy too.