All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Your question included two points:
First: Is Khul‘ an annulment of marriage or a divorce? The preponderant opinion adopted in Islamweb is that Khul‘ counts as a minor irrevocable divorce, not an annulment of marriage. There is another scholarly opinion that Khul‘ counts as an annulment of marriage, not a divorce.
Second: The Wali (bride’s guardian) in the marriage contract. The Wali is a condition for the validity of the marriage contract according to the opinion of the majority of scholars. Abu Haneefah, on the other hand, held that a marriage contract is valid without the Wali. The fact that the Wali merely knows about the marriage and does not object to it is of no consideration.
It is impermissible for a Muslim to choose from the different opinions of scholars based solely on personal preferences or whims, but he should rather choose the preponderant opinion if he has adequate Islamic knowledge or seek the Fatwa of a trustworthy and reliable scholar. If these two options are not available in your situation, we advise you to refer the case to an Islamic center in your country of residence.
Allah Knows best.