Hi Abkyou,
If you look at the first screen shot in my post (for X-Pane 11), you can see that it says, "Finished Web Update". Underneath that it reads, "The technical problem was..", followed by the pathway to the errant file. I followed that pathway and deleted the file listed (pastr_temp_sdry_sflat_d.dds) and ran the update again. That solved the problem I was having. The problem file on your version of X-Plane may well be different but if you have a message like the one shown in my screenshot, you should be able to track it down.
This beta version seems to have a few issues but, no doubt, Laminar are working hard to iron out the bugs! If you continue to have problems, let me know. I'm no expert and am very much a beginner with X-Plane, but I may be able to help.
Cheers