Is what the questioner did—speaking with the girl with the intention of proposing to her, and setting a waiting period before a formal proposal—considered Haram? And should he continue with this step?

It is not permissible to sit alone with a foreign woman (a woman who is not a mahram) unless there is a necessity and a mahram is present. One must adhere to a religious woman and convince the mother to agree to marry her by explaining the Prophet's - peace and blessings be upon him - emphasis on a religious woman. If the mother agrees, then praise be to Allah. If she does not agree, then one must abandon the idea of marrying her, because obedience to the mother takes precedence. The promise of engagement must be fulfilled unless there is an impediment.