Is a betrothal considered legally binding simply by the committed girl hinting at her agreement to marry the committed young man, even if there has been no official announcement through the families?

It is permissible for a man to propose to a mature woman directly, but it is preferable for the proposal to be through her guardians. The decisive factor in accepting the proposal is the word of the mature woman herself if she is not under duress, and the word of the guardian if the woman is under duress. The preponderant opinion is that a mature, sane woman cannot be coerced into marriage. If the girl asks you to propose to her through her guardian, then do so. And know that the proposal is merely a promise of marriage, and the suitor remains a stranger to his fiancée until the marriage contract is concluded.