I think that Observer of times is referring to those who observe future events by looking to the stars, sun and moon. The way of other Nations (Egyptians being one of them) It's like todays psychics and fortune tellers. Those who read zodiac signs. It is mixed in with divination and witches. It is a serious offense to God, so much so, that if someone noticed someone practicing such things, then they was to bring the matter to the Priests and would have to be 2-3 witnesses and if that person was found guilty then the 2-3 witnesses would be the first to throw the stones at them, then the others would throw them at them as well until the wicked people died. The Nation that the Israelites left was Egypt and the Egyptians practiced these things religiously and God did not want His children doing these abominations in the Land that he was giving them.They was getting ready to come into the Land that God gave them and didn't want any corruptness to enter in to it. He just delivered them out of all that.
Deuteronomy 18:9 When thou art come into the land which the Lord thy God giveth thee, thou shalt not learn to do after the abominations of those nations.
Deuteronomy 18:14 For these nations, which thou shalt possess, hearkened unto observers of times, and unto diviners: but as for thee, the Lord thy God hath not suffered thee so to do.
The next verse talks about God raising up a prophet.
You know how people are today, they always want to know the future and know what is going to happen so they can be "ready"
They are fearful of death because they are fearful of hell and should be because that is where they are headed but they want to read all these false prophetic things to ease and comfort them from the fear of hell so they turn to witches, wizards, charmers,diviners because they bring comfort to them and deceive them into thinking that they know what the "future" holds for them and people pay a lot of money to know their future. So I think the observer of times is people who look to the stars to foretell the future.
Deuteronomy 18:20 But the prophet, which shall presume to speak a word in my name, which I have not commanded him to speak, or that shall speak in the name of other gods, even that prophet shall die.
Observer of times is the satan version of Prophet. So I think it means False Prophet
If I am wrong in anything please correct me but this is what I understood of your question about an "observer of times"