Well here is the problem.

The story goes like this:

Two people chat in a bar and when some random song starts to play --
Begin montage
a) another bar - party etc
b) beach - another party
c) club etc - another party
blah blah
End montage.

Notice that all montage scenes include some kind of music, which I don't need to be heard. If I want to have the whole montage 'under' the specific song, where/how do I define that?

Is this formatting correct?
BEGIN MONTAGE (x song plays during montage)

The standard montage will occur under a single song if there is any music at all, so this is what readers would assume. You as the writer would only specify the song if it was important to the story, otherwise the producers will go with something reasonably appropriate and reasonably cheap/easy to license.

I would format a specific song like:

INT. STUDYING MONTAGE - NIGHT

Dana prepares for her exam as "Under Pressure" by Queen plays.

- She pulls thick book after thick book from her backpack onto her dorm room desk

- She pulls a thick book off of a library shelf, through the opening she sees other students goofing off and having fun

- Dana sits on her bed reading, while open books lay everywhere

- Dana struggles to carry several more books through a crowded student center

- She unloads additional books onto her desk, which collapses under the weight

END MONTAGE

Probably, I wasn't very clear...
In order to skip the cost/license issues, let's make the song a piano tone or such.

Allow me to change your example a bit, into this:

INT. DANA'S HOUSE - LIVING ROOM - NIGHT
Dana eyes the piano, sits on the piano bench. Pets its keys, starts to play a familiar tone/some music (This is the tone I want us to listen during the whole montage).
Dana shuts her eyes, continues to play. Everything around her grows darker, so does the music. Into this fantasy flashback--

BEGIN MONTAGE
- She pulls thick book after thick book from her backpack onto her dorm room desk
- She pulls a thick book off of a library shelf, through the opening she sees other students goofing off and having fun
- Dana sits on her bed reading, while open books lay everywhere
- Dana struggles to carry several more books through a crowded student center
- She unloads additional books onto her desk, which collapses under the weight
END MONTAGE

INT. DANA'S HOUSE - LIVING ROOM - NIGHT
Dana takes her hands off the piano. We're back to reality.

So in this case how do you make it crystal clear that the piano tone is heard throughout the montage?

You can do it in the action line, or the acceptable way to do it as a slug is:

MUSIC UP: "Under Pressure" by Queen

MONTAGE

Sally gets dressed.

Sally drives her car across town.

etc.

I suppose your idea would be closer to

INT/EXT. BATMAN UNDER SIEGE MONTAGE - DAY

Bane dismisses several henchmen and sits at a grand pipe organ. He cracks his knuckles and begins to play the classical "Dies Irea."

- A guard opens a heavy door, reveals The Joker in a straightjacket. The guard walks away, and the Joker bolts for freedom.

- Two-Face, in a prison uniform, studies a book in the prison's law library. He opens the next book, and it is hollowed out with a key inside.

- Batman fist-fights The Joker and emerges the victor.

- A heavily-fortified prison transport is T-boned by a cement truck. Croc crawls from the wreckage and runs.

- Batman leans against a wall next to a tied-up Two-Face. Batman wipes his brow.

- The Penguin feeds pigeons from a bench in a prison yard. All of the tower guards take a break at the same time as a large drone swoops down and carries The Penguin from prison.

- Batman and Croc trade blows on the roof of a moving train. Batman knocks out Croc, but is left winded.

- Poison Ivy tends plants in a prison greenhouse. A window slides open by some unseen hand.

- The Penguin gasses Batman with a trick umbrella, but he is able to continue fighting... barely.

- The Riddler sits in the back of a prisoner transport van. Both officers leave at the same time, but not before leaving the keys to his chains and the van on the driver's seat.

END MONTAGE

Bane finishes his musical piece and takes a bow.