The baby has cried four times today.

Your first suggestion is correct.

Thank, InfraBlue.

What about the second suggestion?

The second suggestion makes no sense. If the baby cries after the time of writing, how can that be known?

Thanks, centrox.

That means "The baby has cried four times today" has the same meaning as "The baby cried four times today". Or is there a difference in meaning between the two sentences?

They mean the same thing, but one is more likely in American English.

The baby has cried four times.
The formation [verb "to have" + participle] is the present perfect. He has eaten his breakfast. The dog has barked twice today. I have been to France twice this year. They have heard the music. This tense indicates either that an action was completed (finished or "perfected") at some point in the past or that the action extends to the present (and if we are talking about a present time interval - today, this week, this year).

The baby cried four times.
This is the simple past, sometimes called the preterite, used to talk about a completed action in a time before now. The time of the action can be in the recent past or the distant past.

A British English speaker would be less likely than an American speaker to say "The baby cried four times today" because they would consider the present perfect more appropriate since the focus is on the current day ("today"). They would say "the baby cried four times yesterday" though.

In spoken American English it is very common to use the simple past tense as an alternative in situations where the present perfect would usually have been used in British English. The two situations where this is especially likely are:

(i) In sentences which talk about an action in the past that has an effect in the present:

A = American; B = British:

A: Jenny feels ill. She ate too much.
B: Jenny feels ill. She's eaten too much.
A: I can't find my keys. Did you see them anywhere?
B: I can't find my keys. Have you seen them anywhere?

(ii) In sentences which contain the words already, just or yet:

A:
Are they going to the show tonight?
No. They already saw it.

B:
Are they going to the show tonight?
No. They've already seen it.

A:
Is Samantha here?
No, she just left.

B:
Is Samantha here?
No, she's just left.

A:
Can I borrow your book?
No, I didn't read it yet.

B:
Can I borrow your book?
No, I haven't read it yet.

Thanks, centrox, for such a detailed reply. I appreciate your great effort.