It is important to remember that we are always dealing with real people. Usually these people are not IT specialists, but occasionally IT people have to call for support. I sometimes call a support line for an in-house application and find myself analyzing the performance of the help desk analyst who answers my call.
These days I work in the field. My long years of help desk work are behind me, but even now I think like a help desk analyst and have to stop myself from pointing out their areas of weakness. After all these years I still find that many people start talking too soon when they answer the phone. They also gabble their answering script, mainly because the people who decide what the greeting should be haven't really studied the subject properly.
First, most greetings are too long.
"Good morning! This is the ACME Corporation internal IT Help Desk; my name is John. How may I help you today?" may say all that you want to communicate, but it will be reduced to a meaningless series of noises when the analyst has recited it a thousand times or more.
On the other extreme, the response that I heard last time I called was abrupt to the point of rudeness: "User ID?" was how I was greeted.
I feel that the best response lies somewhere between the two. I think you need to identify yourself and your position but keep it short.
You need to allow the caller to calibrate themselves to your voice. By this I mean you need to give a sample of your voice so that they can learn how to listen to you. Two words is all it takes, so I would suggest that everybody opens the call by saying, "Good Morning" (or afternoon), take a breath, then announce yourself properly, and ask a short question: "This is Jeff speaking, how may I help?" Invite the caller to speak without making them sit through a long, rambling, and meaningless introduction that neither party to the call is the least bit interested in.
When it is necessary to take details from the caller, do it in stages. If you are filling in the call ticket, you don't have to do it all in one go. Start with the name and take the user ID and phone number as you are thinking about the problem. This is a business call, but that doesn't mean that you have to be impersonal. A gram of empathy at an early stage of the call can make the rest of the job a lot easier. Treating the caller like a form that has to be filled in is not the way to do it, although that is, in my experience, the most common way it is done.