I suppose there could be different views. I have not personally encountered the view you describe in the OPC and PCA churches that I've been in. The question is not what practices are out there, but what practice is normative?
What do our Standards say? Anything about being "held in the arms"? No. Here's one example:
28:4 Not only those that do actually profess faith in and obedience unto Christ, but also the infants of one or both believing parents, are to be baptized.
The history of the term "infant" reveals a wide range of description, including well up into childhood. We 20th cent. Americans tend to restrict such a term to children under the age of 2; or less, sometimes less than 1, preferring another descriptive term for older children. But there is no rule to this.
The fact is that it was simply assumed for the Vast Majority of the "Christian world" and "Christian era" (say the past 1500 years) that babes in arms would be persons ordinarily expected at the baptismal font, and only irregularly families with much older children. Primarily in evangelistic territories.
The issue of whom to baptize is theological before practical: was Abraham administering the visible sign of the Covenant of Grace by circumcision? Yes, according to our reading of the text. He circumcised not only his 13 yr old son Ishmael, but ALL the men in his household, including the servants.
Now, I assure you I do not believe Abraham forced his servants into position and threatened them serious damage is they didn't "lie still" for the operation. Most of them were believers already, demonstrably so, having set out from HOME, in Ur, with Abraham leaving everything (family, property) behind, to wander with a man on a mission from God.
And if they had resisted, failed this test of faith that Abraham himself took, there was the door. Here are some nice parting gifts, thanks for coming.
So, the question (as it was presented) seems to assume that a 3, 8, or 11 yr old is capable of responding positively or negatively, in a fully informed way to the claims of the gospel. I think this borders on the absurd--definitely in the case of the 3 yr old, probably in the case of the 8 yr old, and possibly in the case of the 11 yr old, although I know some pretty bright 8-10 yr olds.
Asking a 3, 8, or even 11 yr old of he believes in Jesus, or if he can articulate a gospel-faith, and on the basis of his answers deciding to baptize him--this is simply dumbed-down conversionism, or bringing in the worst of contemporary baptist-belief (note, friends, I am not excoriating baptists there!) into the presbyterian church. And I know plenty of baptists who would blanch if they heard about the sister church up the street baptizing 4-6 yr olds on average. In other words,, both sides should cringe at that example, albeit for different reasons.
We do not baptize minors on the basis of their professions, but their parents'.
Now, we have other issues to deal with. Is the child in subjection to his parents? If the obvious answer is "no" then why should he (or she) be baptized? They are in no frame to be so. If the youth refuses to participate, besides being an instance of insubordination, the church ought not to force it's ordinances on anyone capabable of obvious, willful resistance. And plainly, I'm not talking about a squirming infant here.
I can't tell if a 5 yr old understands the "gospel" I present him with. I can't get inside his head, and he can't explain such things back to me in his own words. But if I can tell Isaac was in subjection to his father when he tied him up and laid him on the altar, then I think I can tell if a child is in subjection to his parents, when they bring him for baptism. At the very least, I have the evidence of my own eyes of a compliant child, and parents who swear he is obedient to them, and that they will continue to discipline him in the faith.
And, we have to deal with the variable age of emancipation. Different ages in different times and places. So, not every child ought to be baptized, nor every person in a modern "household" compared to the ancient. Why? Because some people today may live at home, and be basically there to avoid paying rent, not because he still abides under his parent's authority.
I will not baptize an 18 yr old, not mentally retarded, who only comes to church sporadically, drives his own car, and makes no obvious attempt to unite with the church along with his family. I expect more from him,
because he is an adult for all practical purposes. Wisdom is required in these matters. That is why there is a Session.