Each of the four sentences is grammatical. But there is a difference in meaning between future will and the progressive futurate. (The progressive futurate is the use of the "present progressive" for the future.) You can not use the progressive futurate construction in all instances.
The progressive futurate, according to present theory, means both
(a) the existence of a current plan (we use "plan" for lack of a better word)
and
(b) the speaker is committed to the expectation that the plan will happen.
So, both
1 The meeting will take place on the 5th of June.
and
2 The meeting is taking place on the 5th of June.
are grammatical. Thus, you can use will.
However, in #2 (which uses the progressive futurate) the speaker is committed to the expectation that the plan will happen or be carried out. To see the difference, consider
1b The meeting will take place on the 5th of June but it might not.
This is fine. However,
2b *The meeting is taking place on the 5th of June but it might not.
is not grammatical.
Consider the paraphrase
2c There is a plan for the meeting to take place on the 5th of June, but it might not.
This is fine, but it shows that the progressive futurate itself encodes more than the mere existence of a plan.
Likewise,
3b *I'm seeing the doctor at 5 pm but I might not.
is not grammatical, whereas
3a I will see the doctor at 5 pm but I might not.
is fine.
The classical illustration of this concept is
4 The Red Sox will play the Yankees tomorrow but they might not.
and
5 *The Red Sox are playing the Yankees tomorrow but they might not.
as compared to
6 There is a plan for the Red Sox to play the Yankees tomorrow but they might not.
Number 5 is ungrammatical.
See The Plan’s the Thing: Deconstructing Futurate Meanings (Bridget Copley, Linguistic Inquiry 39(2):261-274, April 2008)(link), who gives a bibliography of studies on the futurate. And also May 2015's Unifying Concurrent and Futurate Progressives by Brandon Robert Beamer (pdf).