xudh-vaqt-am means "My time is passing in a pleasant manner" whilst "xush-baxt-am" means
"My luck/fortune is good". Both mean "I am happy/fine".
True. Also وقت originally has a broader meaning than just "time" in literature and Mythicism it is a concept dealing with the present state of a Person. Most Persian native speakers probably are not aware of this and only use "وقت" in the same meaning as time in general, however unconsciously they differentiate وقت and زمان in usage.
They both mean
time but وقت is usually used in connection with "state of a person in relation with that moment /a certain moment in time" as in وقت ندارم or میشه چند لحظه وقتتون رو بگیرم؟ or رفتم دکتر وقت گرفتم . while in science we strictly talk of زمان (which can be a range of many moments or include minutes/hours etc and can be measured) as in سرعت = مسافت بخش برزمان
Anyway it's a complicated concept and to explain it fully would require pages of writing but so وقت is not exactly just "time" it is the present state/mood of a person, or time in relation to the present mood and how one feels the time and how one is aware of the time etc.
So when we meet someone and want to say "pleased to meet you" it's more accurate to say خوشوقتم which although can literally be translated as "I'm having a good time" but as I explained here "وقت" has a deeper meaning than just "time" in English.
خوشبختم is often used interchangeably with خوشوقتم , its actual meaning is "I'm lucky/fortunate" so if you want to say "I'm fortunate/ lucky because I have a nice family" you can use it : خوشبختم چون خانوادهی خوبی دارم
Both these terms are frequently used, and most people do not consider them much different but as I explained they're different and it's better to use خوشوقتم when you want to say you're having a nice time/mood/ or you're pleased and use خوشبختم where you want to say you feel lucky/fortunate.