This is actually an interesting question. First, though, I don't think "implies" is the right word to use. "Imply" means to say something indirectly but using "today" to mean "today" is pretty direct. Have a look at these two examples:
A. I'll write it today.
B. I'll write it in a few minutes.
Obviously A tells us directly that it will be written today. Sentence B, on the other hand, strongly implies that it will be written today (because "in a few minutes" is unlikely to take us into tomorrow) but it says so indirectly. Is that clear?
So, going back to the original question, in a sense your teacher is right. The sentence "The person writes a letter today" does mean that the person is writing today. However, it probably doesn't mean that the person is writing now. Most likely this sentence implies that the person will write the letter later today. In other words, this usage of the simple present is actually used to show a future action.
Finally, a more natural sentence would be "The person writes
athe letter today." The reason is that it is likely this is a specific person who needs to write a specific letter. The use of the indefinite article "a" suggests that the person is writing an unspecified letter and that would not be so natural.
I hope this helps.
Student or Learner