(Haven't read through the whole thread yet, but wanted to chime in.)
If you really want her to even have a chance of ending the "relationship" you have got to stop caring. By you not being ok with this and punishing her you are pushing her towards him. Now he is the only one who understands her and she can complain to him about how "mean" her mom is. She's thinking you don't understand her and this guy does and he "loves" her, unlike mom who just doesn't get her and tries to stifle her and could never understand this level of love. (In case you're not picking it up, I don't believe that, but she does.) When you get upset at her for having this relationship it upsets her, and guess what, the only person in her life right now that she feels she can talk to is this boy. The relationship would probably not be such a big deal to her if it wasn't such a big deal to you.
I'm speaking from experience here, right after I graduated high school I started dating a guy that was supposed to be just a fun summer relationship before heading up to college, nothing more. Well, my mom had a major issue with it. I wasn't doing anything wrong, I was 18, living on my own, and the relationship was very chaste, so no reason for mom to even be concerned except she hadn't hand picked this guy (throughout high school I was never allowed to date anyone who asked me, my mom would only let me go out with guys she set me up with, yes she is a little psycho

). She was getting so upset and it just drove me to this guy more. I liked him and he was going through her wrath about our relationship with me and therefore was the only one who could really understand what was going on. We got really close and it's because my mom pushed us there, kind of acting like a catalyst for our relationship. So, instead of going out once a week like we had planned before college, I was now talking to him everyday and he ended up going to college with me and now, several years later, we are happily married



.
I'm not saying the 2 situations are the same, I understand your concern over the daughter's lying and her age, but I just thought this might be good food for thought. I don't know if your daughter really is in love or not, but remember that to her it is very real. More than likely the feelings will fade if you just let them burn out. Would it be possible for them to meet and let them go out together? Remember, a big reason why this relationship is working is because it is long distance. Right now they are perfect to each other because they don't have to be around one another. It's easy to love someone when you don't have to be around them when they're cranky or being disgusting. They're idealizing one another because they are not in a situation to discover the other one's faults.
If you decide to just be ok with the relationship and let her know that, there would be no reason for her to be deceitful (not condoning her behavior here). Look on the bright side, you don't have to worry about them being sexually active


. Seriously though, realize that your daughter's at an age where she probably feels that no one understands her. If I were you I would encourage her to have more normal high school experiences and then be ok with her coming home and chatting about them to this guy. But as long as she is viewing you as the "wicked witch" she is just going to burrow deeper into this relationship and it will be that much harder to get her out. Maybe have a conference call or meet with him and his parents and say that you're ok with the relationship as long as it doesn't distract from schoolwork and local friends. Test it out and go from there.
Just my advice from the outside looking in. Of course, the whole topless picture thing throws a whole other wrench in things and I don't know what to tell you, it is completely befuddling to me and seems to be getting more common, probably the result of exposure to things like celebrity sex tapes and whatnot. I understand your frustration and concern, but remember in 3 years she'll be an adult and if she doesn't feel like she can be open with you now imagine how much worse it will be when she's no longer at home. You have a fine line to walk between making sure your daughter is safe and behaving appropriately and not pushing her away and into worse things. I wish you the best
