What is the year of Ethelred the Unready's Laws of London?

tldr: This is usually dated to the 990s.

Unfortunately, it is impossible to answer this with enough precision or certainty to speak of a "correct year". The legal code known as IV Æthelred II (or De Institutis Lundonie, as Benjamin Thorpe titled it) comes down to us via a 12th century Latin compilation, the Quadripartitus, which did not preserve the original's history (if it was even available to the author).

Consequently, any dating is necessarily circumstantial. Consider for instance Felix Liebermann's answer, which as the question mentions placed IV Æthelred between 991 and 1002. Liebermann reached this conclusion based on the apparently friendly relations between the English and the Normans and Danes: 991, when Æthelred signed a treaty with Richard of Normandy; and 1002, when a massacre of the Danes occurred.

However, there are at least two other theories for the possible dating of IV Æthelred II. The first dates it to the currency reform of King Edgar in 973:

Edgar and the c.973 reform would be a plausible point to relocate this code. It is hard to imagine that the most substantial coinage reform undertaken in Anglo-Saxon England was carried out without any written instructions, and “IV Æthelred” is precisely the kind of document that Lafaurie claims was produced for Carolingian reforms . . The case for Edgar’s authorship of “IV Æethelred” is circumstantial and unproven–and likely unverifiable–but it should be considered.

_{O'gorman, Daniel Matteuzzi. "Unius Regulae Ac Unius Patriae: A Standardizing Process in Anglo-Saxon England." Loyola University Chicago, 2015.}

The second is that it dates at least partially to the later reign of King Cnut:

It will be noticed that Section 8. 1 of the Code II Cnut enacts that henceforth the penalty established for false coining shall be amputation of the hand and, as has been pointed out, the penalty before that had been death. There would have been no need for this ' henceforth' if the provisions of Section 5 of IV /Ethelred II had been enacted before II Cnut, whereas IV Æthelred II refers to a decision of the council which provided for the penalty of the losing of a hand.

IV Æthelred 8 refers to both Danes and English, which would be most unusual for Æthelred II but to be expected from a Danish king of England, e.g. Cnut, and, it is found in other enactments of Cnut.

_{Kinsey, Ronald S. “Anglo-Saxon Law and Practice Relating to Mints and Moneyers.” British Numismatic Journal 29 (1958-59): 12-50.}

Regarding the 978 date offered in the question, note that both Brittania.com and Fordham.edu are actually referring to the year Æthelred the Unready's reign began, not the year the law was issued. That is, they are saying, "these are the laws in the time of King Æthelred, who became king in 978". You can see this clearly from the source they are both referencing, where 978 appears only once at:

The Laws of London in the Time of Ethelred . . . Source: Thorpe, Benjamin, Ancient Laws and Institutes of England, pp. 300-303 (Eyre & Spottiswoode, London, 1840). - A.D. 978

_{Cave, Roy Clinton, and Herbert Henry Coulson. A source book for medieval economic history. Biblo & Tannen Publishers, 1965.}

As you see, their source is in term citing a chapter from Thorpe's Ancient Laws and Institutes of England. And indeed, the number "978" does not appear anywhere in that source either except at start of the cited chapter, when Æthelred was noted to have became king that year.

[Ethelred, son of Edgar, succeeded to the throne, on the murder of his brother Edward, in the year 978, and died in 1016. - T.]

_{Thorpe, Benjamin, ed. *Ancient Laws and Institutes of England.*GE Eyre and A. Spottiswoode, printers to the Queen's Most Excellent Majesty, 1840.}

Thus, we're only left with Liebermann's conclusion, i.e. 991-1002.