Update from chat on comments: I want to make clear that I believe you and believe that your assesment that this person may not be telling the whole truth is correct. However since you were not actually guarding the door and they could have said the wrong hour they estimated they rang the bell, there is a tiny small chance you are wrong. In general in interactions, as you said it, it is better to assume good faith, unless you have hard evidence. As I mentioned in comments, It has happened to me people talking right at my face almost and I not noticing. Of course I am a bit of a extreme case, but it is just to give you the picture.
I haven't been in a situation just quite as yours, but I have gotten into other situations were people thought I did something when I didn't or at least they were joking like they thought I did.
First, try to never lie, it always brings trouble. Don't asume someone else is lying either, unless you are sure, some people are just a little awkward and may seem like that they are lying when they are not. Using your words, asume good faith.
In a situation like you were the best position is to tell your truth and accept the other person is telling the truth too. If we asume both were telling the truth a number of things could have happened. For example the person rang incorrectly the bell, I can tell you I have done that many times. Or maybe you were really focusing on something, or in the bathroom or with your headsets and you didn't hear. This doesn't mean you have to doubt yourself, it just means accepting the posiblity that you may be wrong, even if it is, given the circustances, a really really small chance.
If you would have acted like you were telling the truth, and somehow you just missed it, and like the other person was also telling the truth, it would have beem less awkward. When you try to cover for a lie, whether yours or someone elses you give imperceptible hints of culpability. That is probably why your coworkers thought you overslept.
Well that is as a general advice. Now to the specifics you have a couple of options:
- Let it go: If you keep at it, it could make you look even more guilty. Do this and with time people will realize you are not like that at work.
- Deal with small groups of people at a time: If you don't think is viable or advisable to let it go because of the damage to your reputation. You need to implement a "clean reputation plan".
Clean your reputation Plan
You need to talk with people in small groups. Start with the people you are closer with and trust the most. They are going to be joking about it probably. You need to tell them in a honest way that what they are doing is bothering you and it is making you feel uncomfortable and also that you feel it can impact you negatively in the company. Something like:
Look guys, I know I can be a bit of a mess with the alarms and oversleeping, but I am trying really hard to keep a good act at work. I know you are just kidding around but it hurts me that you don't believe me and I also think it could impact my reputation. Please, believe me when I say I was at the office working. I had the music a bit loud because I was alone and I guess that is why I didn't hear the bell, or I don't know, maybe I was in the bathroom or at the phone. But I was in the office working, could we take the jokes down a notch? I don't want my boss to think poorly of me.
The important thing here is to show a serious front, normally when people teases us about our faults we get nervous and laught it off. You shouldn't do that in this case, because they will receive the wrong message. You need to transmit honesty and the concern you are filling. But without attacking them, you don't blame them, you ask them to understand you. If they are your friends and you are nice they will understand.
It is also key that you don't feel like you are trying to cover up for the other person or that you are lying; you are not. You don't know for a fact that the other person was lying, if you act like you are hiding something they won't believe you. It is the whole body language that gets going without even you realizing that you are doing it. I know you are sure this person didn't ring. It would be better if you could doubt it, however if you could just asume good faith, that should be enough to relax your demeanor. As I said it is important to have the right body language, and if you feel you are covering for them, you won't have it.
Then do same with other groups of people you feel you need to clarify the situation with.
Also get together with the intern and tell them that you are sorry you didn't open the door, and tell them how you feel the whole thing is affecting you negatively, and ask this person to give you their support. Like this person could mention that they rang just once, and you were probably in the bathroom or at the phone and you missed it. But tell them he doesn't need to bring it up, just in case someone asks. Otherwise it may look like you bullied this person into covering up for you. Make sure they understand you are not mad or anything.
I know you said they gave you their support already, this is to make it official and make them more inclined to speak up for you, because they see this clearly upset you and also that you are not mad at them.
Next you meet with your suppervisor, or boss. It really depends on the structure of your organization, but I imagine you respond to someone. Setup a short meeting and tell them how this situation has made you worry. Tell him you are working really hard and you were there, you probably just missed the bell because you wearing headsets or because you were in the bathroom when the other person rang, you don't know, but tell him it really worries you that they could think you don't take the job seriously and because they are not around you will slack off. Say something like:
Hey, thanks for this meeting, I have been a bit worried about the
thing of the other day. I really like working in this company and
respect my seniors coworkers, I feel I am learning a lot from them.
That is why I don't want you to think that I am slacking off or not
taking my responsibility seriously just because you were not around. I
am really sorry I didn't open the door to the intern, they must have
rang when I was in the bathroom or when I was wearing headphones. I
promise I will be more careful in the future. I just wanted to let you
know how I feel about the company because I think maybe I came out as
irresponsible the other day.
Basically own the problem, don't talk about the other person, you don't really know what the other person did or didn't do. You just know what they said they do. If you want others to believe what you say, you need to start by believing what others say. If you talk about what the other person did or didn't do it can sound like you are trying to dish the problem on them. At this point you don't care about the intern, you care about you and your reputation. Showing concern for the perception your boss may have of you and taking responsibility for the situation is a way of showing you care.
This has worked for me in the past in similar situations where I was worried about the perception my boss or coworkers had of me. Showing your cards in ernest, telling them what you think and being open to criticism. It is key to accept your responsibilty. And never talk about what the other person did or not do. When you are talking from what you feel about your situation your body language also changes, you are not there to throw the other person under the bus, you are there to tell them what you did and how you feel about the perceived issue.
When people see you are really worried about something, they will take you seriously, and be more inclined to believe you or at the very least put the thing to rest. Because people in general, don't really want to make other people they care about to feel bad.
Finally, be prepared to still be teased about it in the future, when you guys get together for a barbeque or such, it will just be friendly banter, but they won't mean it.