I don't think your transmitter sketch is sending the raw data correctly. The results you got from receiving that output is nothing like what my sketch produced originally.
And actually, what I see in what you got with my sketch is just Sony protocol, 12 bits, with two repeats, each following a 25ms delay. So I think if your sender were sending that, it would work - unless the 40KHz thing is a problem. So I don't think we have the right send sketch. I just don't know what the right one is. Unfortunately I'm still on v2 of IRremote, and have no experience with v4. What is the sketch you used to send? Is it an example?
What do you get with my sketch when you press a number key on the cable remote? Maybe we can figure out what protocol it is.
As for the Adjust setting, we know the Sony values are supposed to be 600 and 1200, so try changing Adjust until you get close to that.
By the way, doesn't your cable remote also handle power on the TV? If so, it should produce the same result as your Sony remote for On/Off. It would be interesting to see if the same Adjust value works for both.