FWIW, I remember from my kids' preschool days that most 4 year olds could only write their first names and that many of them still did that badly, so I think you're right not to put too much worry into this. I also seem to recall many K students unable to write both their first and last name or any discernable letters at all, for that matter. And even quite a few 1st and 2nd graders who made their letters in a weird way even though they ended up looking normal.
Just in case this might work, I thought I'd tell you what I did with my kids that encouraged them to start writing. Thank you notes. Anytime they got a gift, they had to write a short thank you note before they could use it. At first, I would just ask them what they wanted to say and I would write the note with them signing their first name. When they mastered that, I would write the "Dear so and so" part and then draw dots in the shape of "Thank you" which my kids would "connect" to wrtite the words, and then they would write their name. Another technique that worked would be for me to just write "Thank you" on another piece of paper and then have them copy it on the thank you note as best they could from looking at it. Eventually, it progressed to writing the name of the person, thank you, and their name. And by that point they often began to write a short sentence or two. With some of my kids the whole progression was quick, and with others it took a while. After all, there aren't always a ton of reasons to write thank you notes -- usually just birthdays and holidays. Obviously, you could do the same thing with any kind of correspondence, but we found it worked with thank yous because we could emphasize how nice it was for someone to give them a gift and how it was important to say thank you before using the gift. And besides, sometimes they just wanted to use the gift!
Anyway, I could see writing down their phone number in dots and having them connect the dots, which gets them writing and learning their phone number. It would work also for their address, their birthday and Christmas wish lists, etc.
BTW: I totally get the attitude of not wanting to be the student. I never wanted (or want) to look like I don't know what I'm doing. Heck, when I was a kid I didn't even want to play in the front yard for fear of our neighbors or even passers-by thinking that I was childlike!
