Past Simple vs Present Perfect for not finished period of time in AmE

TL;DR

You can say:

I drank three cups of coffee this morning [or "today"].

You would use this sentence is you perceive the action of drinking coffee to be completed or if the morning has passed (if you choose "this morning" instead of "today")

OR

I have drunk three cups of coffee this morning [or "today"].

You would use this sentence is you perceive the action of drinking coffee to be still occurring.

Whether you perceive the action as being "completed" or "still occurring" is your choice.

---

There are a few things we must consider before approaching this.

First, we need to choose between "this morning" and "today." I will address both of these in my answer.

Second, we must know what time of day it is; this tells us if it is still morning, or if the morning is in the past. I will address both situations in my answer.

---

The simple past (or past indicative) is typically used to talk about things that happened (and were completed) at a specific point in the past.

Note: the author/speaker can decide what constitutes "completed."

For example, if it is either still the morning or no longer the morning, we can say the following:

I drank three cups of coffee this morning.

Regardless of whether the morning has passed, we can use the simple past, but only if you perceive the action of drinking coffee to be complete. Then, the action (drinking coffee) happened at a specific point in the past (this morning), so the use of the simple past is warranted.

You can also use the simple past with "today" instead of "this morning."

I drank three cups of coffee today.

As I stated above (when "this morning" was in the sentence), you can only use this tense if you are not going to drink any more coffee today. This means that the action has been completed at a specific point in the past (today–but before now), which warrants the use of the simple past.

---

The present perfect is typically used to describe actions that began in the past and are still occurring. The present perfect is formed with the verb "have" or "has" and the past participle of the verb.

Note: the author/speaker can decide what constitutes "still occurring."

For example:

I have drunk three cups of coffee today.

The action (drinking coffee) began sometime today and is still occurring. It is still occurring, so we don't want to say that it is completed (which would make us use the simple past).

If it is still the morning, we can use the present perfect:

I have drunk three cups of coffee this morning.

However, you should only use this tense if you do not perceive the action (drinking coffee) as being completed.