I personally don't think it's cockiness or bad habits that causes prior civvy guys to have problems. Yeah there are some lowest common denominators out there that those things apply to, but I think the biggest thing is just an inability to adapt to military flying. It's very different, and everything that was once the hardest part of your flight had to now become like breathing......nobody expects you to fight a 2V1 BFM and then airnav back 120 miles to home plate as a wingman and land your jet at absolute single pilot mins on min fuel when you first start out, but that kind of stuff will be expected in a year or two and a couple hundred hours later. I think the guys with prior time that struggled were the guys who just couldn't drink from the firehose for whatever reason
THIS.
When I was teaching tweets and T-6's about 2-3 years ago, most of the guys who had significant prior time (everyone had at least IFS/IFT), did very VERY well. There were about 1 or 2 counter-examples, though. One of the guys who just sank straight to the bottom of his class despite his prior experience was just a bone-head. Not high on the common-sense scale, and not particularly book smart either. For that reason, he started out the first 5 or 6 flights ahead of his classmates... but they caught up and passed him in very short order. I don't know what he, or anyone else, could have done to make his experience go any better.
The other prior experience/airline guy that didn't do very well that I had experience with personally wasn't stupid... he was stubborn. There is some variance from person to person and background to background in how you teach the flying gig, and Air Force instructors do tend to differ from civilian instructors in some areas. For example, most Air Force guys tend to teach pitch for altitude and power for airspeed, and I gather that most civilians tend to teach the opposite (I've been taught by basically all USAF with some very small exceptions here and there-- so I can't say even from my own experience what methods "most" civilian CFI's teach). Most USAF guys, or virtually all of them really, teach the control/performance concept for instruments rather than primary/supporting. And of course the methods that the Air Force uses airplanes differs from the civilian world. The problem set can be different (depending on the airframe), so of course our solutions can also be different. Et cetera, et cetera.
As instructors, it is incumbent upon us to develop several different teaching styles and adapt them to the student... but as a student it can be even more important to try to learn from instructors who have different teaching styles or techniques than those with which you are familiar or comfortable. After all, as a student you need to learn the material more desperately than the instructor needs to teach it. If the student fails to learn... the student washes out. If the instructor fails to teach... the student washes out.
Student #2 failed to adapt, or to be adaptable, really. To be fair, his instructors were hardly god's gift to aviation... or god's gift to instruction for that matter. But they were all at least competent and adequate (and many were really quite good), and they were teaching him techniques that were successful for the environment that he was going to be operating in. And they ran the whole range in terms of personality type, from buddy-buddy nice guys to authoritarian hard-ass types. From technically-minded aero-engineering math geeks to the "keep it simple, stupid" history major types. He seemed to be unwilling to accept any of them, however. He had "great" reasons for why his way was "better," ... except for the fact that his maneuvers didn't match the required parameters. I wouldn't have classified his problems as cockiness or bad habits either. I don't even think he was being intentionally stubborn. He just had an unwillingness to give up a prior mind-set for a new one... so rather than being a sponge like his more "blank slate" classmates, with him, it seemed like a lot of it just didn't "stick."
Neither of these guys washed out, but they did bump along the bottom of the class until track select getting close to washing out on several occasions.
As for what to tell your instructors or fellow students about your background. Tell the truth. Lying is unbecoming of a military officer and against the UCMJ. Your background is your background, and any negative light you seek to shield yourself from by hiding it is going to be twice as bad if people think you are trying to hide something. On the other hand, don't advertise it either... and definitely don't try to use it to impress anyone, including (and especially) yourself. You're there to learn, just like everyone else. That's the attitude to take, IMHO.