subir is a word that is difficult; its several meanings make it hard for me to select the right one.
On reflection, I agree with you.

In the translations of the folk song I found it has the sense of 'to come up', or 'to enter' (given as one of the definitions in the dictionaries I looked into).

'To mount or to climb' is the common definition and why the other is used in the translations I found, I don't know.

Also pino verde suggests a single tree, not a wood.