Theoi.com has two Pages on Perseus. On the second Page, the last chapter (which is followed by some appendices) is entitled "The Death of Perseus". It is very brief, listing the only two characters involved, aside from Perseus, and these are Proitos [Proetus] and Megapenthes, which listing is followed by this blurb:
The only reference to Perseus' death is a very obscure legend
recounted by Hyginus.
After that it quotes from the relevant portion of Mary Grant's 1960 English translation of Fabulae 244, which originally is entitled Qui cognatos suos occiderunt, "Men Who Killed [Their] Relatives," and is basically a quick roll-call of fifteen culprits who fit this bill.
Right in the middle of the list:
Megapenthes Proeti filius Perseum Iovis et Danaes filium, propter
which is translated by R. Scott Smith & Stephen M. Trzaskoma1 as the name of the killer followed by that of his victim:
Megapenthes son of Proetus: Perseus, son of Jupiter and Danae, in
revenge for his father's death.
And that is all that Hyginus has to say on the matter, which brevity is probably what Theoi means when it calls the story "obscure".
In the sentence immediately after what you have quoted from William Smith's Dictionary of Greek and Roman Biography and Mythology, it says that the scholion on Euripides' Phoinissai 1109 describes Proitos as having been expelled from Argolis (presumably by Perseus) and having ended up in Thebes.
This appears to be a different version of Proitos' fate from the one in which his great-nephew Perseus kills him, which then would preclude his son needing to avenge his death because Perseus isn't responsible for that in the scholion.
But then the scholion doesn't offer thereafter an account of Perseus' death. Strangely enough, though, Hyginus, earlier in the Fabulae, has already enumerated Perseus in his list of sixteen "Mortals Who Were Made Immortal" as his Chapter 224 is entitled. The fourth profile on this list reads:
Perseus, the son of Jove and Danae, who was admitted into the stars.
This need not be a contradiction, however, as several others who were made into constellations, like Perseus was, were originally mortal characters who died first before being lifted into the skies (e.g. the giant Orion, the Hydra, and the sea-monster which Perseus killed in order to save Andromeda [the last of these becoming the constellation Cetus]).
Malalas' Khronographia [Chronographia]
Later still than Hyginus, the Early Mediaeval Antiochene writer John Malalas (circa AD 491 – 578) compiled a history of the world featuring a mixture of euhemerised Greco-Roman myths, snippets of allegory, and philosophical maxims and musings, interacting with characters from the Bible, and entitled the Khronographia, "Chronicle." His treatment of the Perseus myth seems to be unique to him, and his Perseus is somewhat of an evil wizard supervillain.
Khronographia 2.14 tells of how Medusa, rather than being a monster, was merely a wild-haired, wild-eyed country girl who Perseus, completely unprovoked, beheaded by the roadside in Libya. Using "the deception of loathsome sorcery" that he had learned from his father Picus (here a euhemerised form of Zeus), he "performed mysteries over" the girl's severed head and thus transformed it into the petrifying weapon for which it would gain fame and by which he was able to subdue his enemies.
In 2.15, from Libya, Perseus goes on to Ethiopia, whose virgin princess Andromeda has been dedicated by her father King Kepheus [Cepheus] to serve in the local temple of Poseidon, where she now lives. On account of her great beauty, Perseus ravishes her away from there, making her his wife.
After several conquests all over the Near and Middle East, Perseus comes to rule over Assyria and Persia. In his book Perseus (pp. 32-33), Daniel Ogden2 discusses the end of the character's life as narrated by Malalas in Khronographia 2.21:
Two accounts of Perseus' death survive, but neither was canonical. As
we have seen, Hyginus preserves the bare information, probably derived
from an anomalous tragedy, that he was killed by his
cousin-once-removed Megapenthes in revenge for the killing of his
father Proetus (Fabulae 244). Rather more interesting is the account
of Perseus' death given by the fixth-sixth-century AD Christian
chronographer John Malalas:
After sometime King Cepheus, the father of Andromeda, came
against him from Ethiopia, and made war upon him. Cepheus was unable
to see because of old age. Perseus, hearing that he was making war on
him, became very angry and went out against him brandishing the head,
and he showed it to him. Because he was unable to see, Cepheus rode
against him on his horse. Perseus did not realize that he could not
see, and reasoned that the head of the Gorgon he held was no longer
working. So he turned it towards himself and looked at it. He was
blinded and frozen like a corpse and killed.
John Malalas pp. 38-9 Dindorf (cf. George Cedrenus 1.41)
It is difficult to gauge the tone of this story. Is it tragic? Or are
we rather to laugh and visualise the action along the lines of an
Oliver Hardy peering down a hose-pipe to see why the water isn't
coming out? Is it a Christian joke at the expense of one of the
principal pagan heroes? But such a tale may also have originated in a
pagan joke. A favourite theme of the Perseus tradition from the
second-century AD Pseudo-Lycophron onwards was that Perseus created
statues with the Gorgon-head...
1. Smith, R.S., & Trzaskoma, S.M. 2007. Apollodorus' Library and Hyginus' Fabulae: Two Handbooks of Greek Mythology,
Translated. Hackett Publishing Company, Inc., Indianapolis/Cambridge
2. Ogden, D. 2008. Perseus, Routledge: Taylor & Francis Group, London and New York.