I usually answer on the phone "First name, specialty intern" to other docs. I like other docs to know up front that I'm an intern, that way they know why I sound dumb. I noticed the consultants were a lot nicer to me once I lowered their expectations. You only have the target on your back / shield of being an intern once, might as well use that to what little advantage it offers, namely the only advantage which is lowering expectations. Sometimes you're taken less seriously/discounted/talked down to as a result, but I'd rather that then have any misunderstandings about how much weight I have on the team, and at the end of the day, if they think less of me for being an intern, have at it. Human punching bag at your service if that makes you feel better about your day oh mighty one above me in the hierarchy.
That can backfire with nurses or other ancillary staff calling you, they can get some attitude that you're an intern and they know better than you (and I'm not going to say that isn't the case sometimes) but it doesn't matter as much once I write the order, because an order is an order. They may try to go over your head to the resident or attending, but they usually get the smack down for that. "Sorry, you'll need to page/talk to the intern." Sometimes that leads to a question like, "Intern, why am I getting your pages/having whoever bother me?" Then I just tell them that I talked to whoever it is and they didn't seem to like the fact that I'm an intern. Usually dissipates the tension.
I never figured out if I should use my first name or last with the nurses, I started off with doctor last name so they would recognize the orders that I write and that I'm someone who can write them orders. It seems like most of the other resident docs use their last names and get more respect, but some nurses seemed to think I was the biggest a-hole ever, so when I introduce myself to them now I just use first name last name specialty resident to split the difference. As an intern you're entitled to call yourself a resident, I use intern with other docs to try to soften the blows out of pity, although sometimes they just hit harder because some people are hyenas and they like to go for the weakest in the pack. After that intro, I tell the nurses to call me by my first name and I feel like we get along better for it despite what I said above about respect. I'll go for inspiring likeability over authority with nurses any day at this point. If a nurse calls me doctor last name, I don't correct them because if that's how they think of me so be it. I can use every ounce of confidence in me that you want to throw my way.
Throwing in your status as intern, resident, whatever, lets people know your place in the hierarchy (ie doctor) without throwing your title around.
If I'm calling outside the institution, like to another doc's office, if it's the office staff I say doctor last name because I want them to know to patch me through, then once I get the doc I use first name, specialty intern.
Sometimes calling within the institution I use doctor last name, like if I'm trying to get through on nights to a human being on pharm or lab or something. Same thing with the operator. Not to throw it around but so they know doctor last name means business about getting a hold of someone.
This probably seems like I'm overthinking it, but I'm very psych-minded and I tell you I can feel these little nuances in my interactions with people.
I knew an intern that was using first name with patients, and I gotta say I think it's a no-no. Having the patient call you doctor is not for you and your ego, it's for the benefit of the patient and the therapeutic alliance, if you ask me. They want to have the attention of a doctor, and you want their respect to translate to hearing your advice.