I know some Jews that try to drink most of the cup which they used for kiddush, havdolo, birchas hamozoin, bris, or wedding, (even when the cup is big)
Is there source for what they are doing?
Or is it a misconception from Passover (harav 472.19)?
See the answer to the question here, it appears "Rov Kos" is only a consideration at the Pesach Seder, the rest of the year the amount need to drink may either be a sip, cheekful, or majority of a Revi'it.
Shulchan Aruch OC 271, 13: One should drink one cheek, this quantity is equivalent to the majority of a Reviit for a man of average size. The Ran said that even nobody needs more than a Reviit.
In siman 472, 9 he said, following the comment of Magen Avraham that he needs to drink the whole cup lechatechilla and Bediavad the majority of the cup.
The Bet Haddash on the Tur at the same siman said that the difference between the kiddush and the four cups of Pesach is that at Pesach we have a special requirement of "cups" which refers to a whole cup. For this reason the Ramban stated that even if the cup is big and contains several reviiyot, the din in Pesach is the same, to drink the whole cup lechatechilla and bediavad the majority. The Beur Halacha on Siman 271: adds a proof from the Shulchan Aruch in the same siman. For Kiddush SA wrote melo lugmav in a lechatechilla statement. Indeed, in Gemara Pesachim 107a melo lugmav is mentioned in bedivad sentence. Beur Halacha explains that the intent of SA is to explain that there is no din of full cup or majority of cup lecatechlla
For Kiddush the needed quantity is Reviit and according to some poskim the reviit can be drunk by several persons.
The person making Kiddush for Shabbos (Friday night) only needs to drink the shiur. Other uses of the cup also involve the shiur. However, since the becher is usually small enough, then it is a "rov kos". Since many bechers are of the size for which the shiur is a rov kos, people would tend to regard it as easier to always do that rather than measure. Alternatively, many bechers are designed to be used for the entire family. That is, a person pours out much of the wine to be distributed to the family and only drinks the remainder. Here too, people tend to think of what they drink as "rov kos" because it is the rov of what is left after what has been distributed.
The minimum size for a kiddush cup is 86cc (2.91 fl oz). For Friday Night kiddush, some poskim write that one should use the larger cup of 150cc (5.07 fl oz), which is the Chazon Ish shiur (Biur Halachah to 271:13 writes that for Torah mitzvos one should use the larger shiur, and adds that although kiddush on a cup is not a Torah mitzvah, because the actual kiddush is de’oraisa, it is proper to use the larger shiur, or at least two kebeitzim, which works out to 115 cc or 3.9 fl oz). Others rely on the smaller shiur even for the Friday Night kiddush, citing proofs that this is the ‘true shiur,’ and that this is the custom.
You should use a special kiddush cup that holds at least 4 1/2 ounces. Fill the cup to the rim (our joy should be "full"). If you don't have a kiddush cup, any cup many be used, as long as it holds 4 1/2 ounces and is not disposable.
When the blessings are complete, everyone should be seated (if they weren't already). The one who led the blessings then drinks at least 2 ounces of the wine or grape juice in one or two gulps (this is no time to savor and sip). "Drinking," according to the Talmud, constitutes at least a "cheekful," or approximately two ounces. Only the person making kiddush on behalf of everyone present need do this.
The balance of the kiddush wine or grape juice is then distributed in little glasses, or cups, to all those who were included in kiddush (they need have only a taste).