You are correct that 食べる is a durative verb and that 落ちる and 死んでいる are instantaneous verbs. That’s why 落ちる and 死ぬ were used as examples; the have to show resultant state through ている. As a durative verb, 食べる and do both. In fact, the paper uses 食べる on pages 19 and 20 to explain the difference in how they function.
Transitivity is the reason why I consider 食べる a weird word to use for this discussion, as it is both transitive and intransitive in English. The OP’s question included “I already ate,” which is the intransitive use of “to eat,” which already puts nails on the road to Japanese translation. 食べる can function syntactically as an intransitive verb, but it is grammatically and semantically a transitive verb.
Contextually, there can be a big difference between the following statements:
“I already ate.” This is what I believe @Leebo and I are mainly arguing to be 食べている.
“I already ate it.” This I think would be more natural as たべた or たべちゃった.
In the first, you’re declaring a state lacking of hunger. It doesn’t matter what or when you ate. In the second, you’re declaring that at some point, you ate something.
I won’t vouch for @Leebo, but my reasoning for assuming 食べた would be an recent action is because of its combination with “already” or もう.
Asking, “Did you eat?” 「食べたのか？」, provides no time span. But adding もう (“already”) assumes a time span; a relatively recent one, regardless of if you use 過去形 or 現在完了形. But because 過去形 indicates the time of the action, while 現在完了形 indicates the time of completion of the action, attaching もう to each pulls the time of the action itself closer (as in the time in which you were eating, not when you finished, is closer) when using 過去形 while it pulls the time of completion (not when the action occurred) when using 現在完了形.
“I ate” involves the entire process: sitting down to eat, eating, and finishing. “I have eaten” involves the point at which one finished. This is displayed by how we attach temporal phrases to these statements (as I said before, depending on your region, both may be acceptable for this, but only one is proper in formal English).
“I ate at 7” likely means I started at 7, but could also mean we were eating for the rest of the night. “I have eaten at 7” means we finished eating by 7 and did something else afterward. We have no way of knowing when they started.
Adding “already” emphasizes why “to eat,” a durative verb, cannot be in simple with a temporal adverb.
“I already ate at 7” is used in American English but is grammatically incorrect. Why? Eating is a process that takes a duration. Stating that you “already ate” indicates that, in one minute, you began eating, consumed the food, and finished eating. This is why we often say “I already finished eating” (because “finish” is an instantaneous verb when used intransitively and participles function as independent timeframes).
On the other hand, “I have already eaten at 7” states that one finished eating at 7. As completion can exist in an instant, this properly illustrates what occurred and also explains why the simple past would indicate a more recent development.
EDIT: Speaking with one of my JTEs (my previous consultants were my 国語 teachers), her first reply was that 食べている indicates when one starts eating, similar to the comment you recieved. As we talked about it using the English, she realized what I meant. The overarching problem here seems to be transitivity. Her conclusion is that 食べた is more suitable if you are using 食べる transitively while 食べている would be more suitable when you are using it intransitively (just discussing eating or meals in general). Similarly to American English (as above with the formally incorrect “I already ate”), she said 食べた is probably more common, especially with children, as its the grammatically easier to use and remember of the two.
All I can say in the midst of this: holy crap, language is difficult and interesting. 
