Usage of "to coax"

[Chicken Sandwich]

Isn't this the point where the world-weary barkeep absentmindedly wipes down the bar and coaxes the woes
out of the troubled patron?

Coax means, according to the Longman dictionary of contemporary English:

coax / kəʊks $ koʊks / verb [ transitive ]

1 to persuade someone to do something that they do not want to do by talking to them in a kind, gentle, and patient way : ‘Please, Vic, come with us,’ Nancy coaxed.
coax somebody into/out of (doing) something We had to coax Alan into going to school.

coax somebody to do something We watched the bear coax its cubs to enter the water.

coax somebody down/out/back etc Firefighters managed to coax the man down from the roof.


2 to make something such as a machine do something by dealing with it in a slow, patient, and careful way coax something out of/from/into etc something He coaxed a fire out of some dry grass and twigs.
The driver coaxed his bus through the snow.


— coaxing noun [ uncountable ] : She needs a bit of gentle coaxing.

— coaxingly adverb

In the sentence I quoted above, there is no persuation, so definition 1 doesn't seem to apply to the sentence.

I'm guessing that "to coax the woes out of the troubled patron" means to make the woes go away or persuade the patron to share the woes with the barkeeper.

Does anyone know if I'm right? Thanks in advance.

 

[Tdol]

The barkeep gets things going- if they didn't encourage them to start taking, the patron wouldn't talk about their problems. There is a form of persuasion- the patron would remain silent and the barkeep gets them to talk and unburden.