HAMLET: WHICH HEROD, WHICH BAPTISTA? (JOHN the BAPTIST?)

In Shakespeare’s Hamlet, when the prince advises the players against over-acting and says it “out-Herods Herod,” which Herod is he referring to?
—Herod I who ordered the slaughter of male children in the gospel tale of the wise men and the birth of Jesus (Mt 2:16-18)? This makes sense, but is that all?
—Or Herod Antipas, who said he’d give his stepdaughter (Salome, by tradition) anything she wished, up to half of his kingdom, after she danced pleasingly for him and his guests (Mk 6:21–29)?
—(Or both?)

Later, when the players are acting out The Murder of Gonzago, why does Hamlet say the murdered duke’s wife is named Baptista? Merely because an Italian Duke of Urbino was murdered by poison, and one of his historical predecessors had a wife named Baptista (a stretch)?

Or is the common thread the fact that Herod Antipas granted Salome’s wish for the head of John the Baptist (to whom the name "Baptista" points), who condemned the marriage of Herod Antipas to his brother’s wife?
—In Hamlet, Claudius also marries his brother's wife, and Hamlet, like John the Baptist, disapproves.

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Update 20 December 2021
Song Cho has an excellent article from November of 2014 on Herod, John the Baptist, and Salome echoes in Hamlet. You can find it here:
http://www.languageinindia.com/nov2014/songchohamlet2.pdf
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UPDATE: 22 November, 2020:
See also  John Gillies
Shakespeare Quarterly
Johns Hopkins University Press
Volume 64, Number 4, Winter 2013
pp. 396-424
10.1353/shq.2013.0057

Although Gillies doesn't make the connections in the quote below that I do
- between Salome and Laertes/Ophelia,
- or between Claudius and Herod Antipas who married a brother's wife,
- or between Baptista, the player queen, and John the Baptist who condemned Herod Antipas' "incestuous" marriage to a brother's wife,
- or between John the Baptist and "prophetic soul" Hamlet,
Gillies does discuss Herod (Antipas), Salome, and John the Baptist in the context of Hamlet:

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Link to my new article exploring this topic:
https://pauladrianfried.blogspot.com/2018/11/hamlet-footnotes-in-need-of-updates.html

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[Originally posted around the week of 11/5/18
on LinkedIn]
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Links to a description of my book project:
On LinkedIn: https://lnkd.in/eJGBtqV
On this blog: https://pauladrianfried.blogspot.com/2017/05/hamlets-bible-my-book-project-im.html
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