"I turn 31 tomorrow" vs "I’m going to turn 31 tomorrow," vs "I’ll turn 31 tomorrow"

As I'm guessing you already know, the present tense can refer to the future.
English present tense

Doing a Google search of the variations, I get:

"I'm going to turn 18 tomorrow" = 9 results
"I'll turn 18 tomorrow" = 122 results
"I turn 18 tomorrow" = 8,480 results

There's another one I would tend to use:

"I'm turning 18 tomorrow." = 3,600 results

As to what the differences are, well I think any differences, other than that they use different words and sound different, are unremarkable or even non-existent. There are the very slightest of differences between "will" and "going to", and the following links try to explain some of them.

Will vs going
Will vs going to
Check out this chart
These aren't hard and fast rules.

The nuances among each version mentioned in the The Atlantic article may be referring to nuances of meaning, in which case I don't consciously know what they are, or it could simply be referring to the nuances of speaking different words and the resultant effect it has on a listener.

I really do think they all mean the same thing, and if there are any nuances among them, I'm not really aware of what they are.