Alternate names and birth order.

[Ghostwheel]

I've read that sometimes certain names would be favored in certain families.  To reduce confusion, when a father had one form, he would give his son a different form of the same name.  Forms sometimes being very different from what we would conceive of as being the same name.

For some examples see: (though I am sure there are others)
https://www.libraryireland.com/matheson/varieties/different-names-interchangeably.php

Has anyone seen one of these names alternating between generations?  And more to my question: what did the birth order look like?  (I've never seen one with an intact family record.)

I know naming patterns were often not followed to the letter, but I was wondering if you would still generally expect a son with a different form of his father's name to be born later in the birth order of sons.  Like, generally, third or fourth, instead of first or second.

[aghadowey]

I've read that sometimes certain names would be favored in certain families.  To reduce confusion, when a father had one form, he would give his son a different form of the same name.  Forms sometimes being very different from what we would conceive of as being the same name.

Certainly some families used nicknames for children when they had the same name as a parent but I've not come across what you are suggesting and neither does the article say this.
What the article is talking about are variations of the same name, such as Edward/Edmund where the same person might be known as either name depending on who recorded the information.
In Ireland there are certainly lots more variations- not just for the few names mentioned but more variations of the ones mentioned. For example, Johanna/Hannah/Anna or Peter/Patrick.

[Ghostwheel]

Like here, on p29, when it talks about a father being called "Mago" and the son being called "Mane."
https://archive.org/details/varietiessynony00math/page/n32/mode/2up

I assume that the same phenomenon existed with other variants that is why I referenced the list.

I've found this census record with a father and son "Mago" combo in Dingle:
http://www.census.nationalarchives.ie/pages/1911/Kerry/Glin/Ballintaggart/105872/

On the birth cert, though, it just seems to give the same form for both: "Meago."
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1908/01629/1652046.pdf

A marriage cert with two "Magos": (same form)
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1891/10664/5885533.pdf

I'm hoping to find something where the two forms are different.  Something that would give me a clue as to birth order.

[Maiden Stone]

Like here, on p29, when it talks about a father being called "Mago" and the son being called "Mane."
https://archive.org/details/varietiessynony00math/page/n32/mode/2up

I assume that the same phenomenon existed with other variants that is why I referenced the list

The book was written by the Registrar General of Ireland in 1900. Those examples you quote from page 29 were particular to a few named districts.
Matheson cited examples of other non-standard information on registrations which had been reported to him by his staff,  such as widows in some districts reverting to their maiden surnames and children in some districts sometimes being known by their mother's maiden surname. Those were peculiarities which had been noticed by registrars in some districts but may not have been widespread. 2 common things in my family's county of Mayo were Augustine shortened to Austin and Dominic spelled Dominick.
You seem to be trying to create a general rule from specific, local examples.