The Theoi Project website 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 The Theoi Project 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 from 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]).
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