We just went through this with some other guy. I have a hard time understanding how so many people seem to get a DUI but somehow find a way out of it. You either did it or you didn't, which one? If you are successful in getting it reduced to a wreckless than that is what your DL record will show. HOWEVER!!! A criminal background check will show the original offense and then what it was reduced to, so, DUI will be able to be found. In my opinion, and that is all that it is, just an opinion, if you are asked about it you should explain the whole situation and why it was reduced to wreckless. I don't know if you are asking this question in relation to how to handle it in an interview or on paperwork but, if you choose to disclose what really happened and only say that you have a wreckless driving charge, when the truth comes out you WILL look like you were trying to hide it. The bad thing is that most rational people will wonder what else you are hiding. Face it, you messed up, we all make mistakes but do yourself a favor and take responsibility for it and accept it. Just because you find a loophole in the system or have a good lawyer does not mean it never happened, cops don't just give DUI's out for fun, they are usually rightly deserved. I am in no way trying to be crude or sarcastic because I understand your situation, I am just telling you the cold hard truth.
P.S. The reason I know so much on the subject of background checks is because I just completed 2 weeks of researching my own. Believe me, you would be suprised to see what can come up. My final thought is, if it was that easy for me to find it with a few bucks and the internet, anyone can find it.
Good Luck, God Bless..