Current mainstream Christian and Rabbinical traditions choose option 1 instead of option 2.
Thus there were giants in those days (before the flood) and also afterwards (after the flood). The giants after the flood were descended from the giants before the flood (a slight point in favor of the flood not being global). Goliath and his odd kinsmen were examples of such giants, and it was reported by the spies that there were giants in Palestine, as per Numbers 13.33:
There we saw the giants (the descendants of Anak came from the
giants); and we were like grasshoppers in our own sight, and so we
were in their sight.” [NKJV]
However, some rabbinical traditions at the time of Christ that were popular in Palestine choose option 2. Option 2 does not speak to the universality of the flood at all, as it would allow for more such offspring even after the flood. That is, many believed there were still libidinous angels that would seek out virgins to sleep with or at least violate somehow. We also see this in Paul's writings:
1 Cor 11.10:
Because of this, the woman ought to have a symbol of authority on her
head, on account of the angels. [LEB]
The interpretation of the phrase because of the angels will determine whether option 1 or option 2 is believed at the time of Paul's writings.
One possibility of the head covering was to ward off these angels. Here is the (excellent) Hermeneia commentary discussing interpretations of this passage[1]:
What is the meaning of διὰ τοὺς ἀγγέους, “because of the angels”?
Several suggestions are made. The fallen angels of Gen 6:1f. are
meant. The demons are held to be sexually libidinous. Yet the
thought need not be that of their sexual desire in particular. It can
also be a general allusion to the possibility of woman in her weakness
being harmed by demons. Others think of the order of creation,
arguing that the angels are the protectors of this order. But
Lietzmann rightly objects that in that case ἐξουσία would have to be
understood as a sign of subordination, which does not suit the word at
all. Finally, there are those who think of the presence of angels at
divine worship. If the statement is understood in its context as
giving a reason (διὰ τοῦτο, “for this reason,” and διὰ τοὺς ἀγγέλους,
“because of the angels”), then the ἐξουσία is a protection, in the
sense of a compensation for the natural weakness of woman (in
metaphysical terms: because she is God’s image only in a derivative
sense) over against cosmic power. Paul has no interest in further,
more precise definitions.
This connection was also argued by Tertullian[2]:
Of course, it is on account of the angels, he says, that the woman’s
head is to be covered, because the angels revolted from God on account
of the daughters of men. Who, then, would contend that it is only
women, that is, married women no longer virgins, that are a source of
temptation? Unless, of course, unmarried women may not present an
attractive appearance and find their lovers? Rather, let us see
whether it was virgins alone whom they desired when Scripture speaks
of the ‘daughters of men’; for it could have used the terms ‘men’s
wives’ or ‘women’ indifferently. But, since it says: ‘And they
took to themselves wives,’ it does so because they took as their wives
those without husbands. Scripture would have used a different
expression for those who had husbands. Now, they could be without
husbands either because they were widows or virgins. So, in naming the
sex in general by the term ‘daughters,’ he embraced species in genus.
Thus this interpretation would support option 2. One can also view this prophetically, e.g. that the bride (the church) is subject to being defiled (committing fornication or adultery) by evil spirits when she is not under authority, and the offspring of such a union is a grotesque monster, a mixing of the church with the world.
However others argue that "because of the angels" in Paul's writing is to be taken that we have guardian angels that protect us, contrasting Gen 6.4, so even though this passage is viewed as a reference to Gen 6.4, the meaning is reversed. For example, St. John Chrysostom [3]:
Hence it is evident, that the saints have angels, or even all men. For
the apostle too saith of the woman, “That she ought to have power on
her head because of the angels.” And Moses, “He set the bounds of the
nations according to the number of the angels2 of God.” But here He is
discoursing not of angels only, but rather of angels that are greater
than others. But when He saith, “The face of my Father,” He means
nothing else than their fuller confidence, and their great honor.
And this became the more widely accepted tradition in modern Christianity, which would support option 1 rather than option 2.
[1] Conzelmann, H. (1975). 1 Corinthians: a commentary on the First Epistle to the Corinthians (pp. 189–190). Philadelphia: Fortress Press.
[2] Tertullian. (1959). Disciplinary, Moral, and Ascetical Works. (H. Dressler, Ed., R. Arbesmann, E. J. Daly, & E. A. Quain, Trans.) (Vol. 40, pp. 178–179). Washington, DC: The Catholic University of America Press.
[3] John Chrysostom. (1888). Homilies of St. John Chrysostom, Archbishop of Constantinople on the Gospel according to St. Matthew. In P. Schaff (Ed.), G. Prevost & M. B. Riddle (Trans.), Saint Chrysostom: Homilies on the Gospel of Saint Matthew (Vol. 10, p. 368). New York: Christian Literature Company.
