Not being an expert in such cases, here are some thoughts & clues.

There are children named with 7 different surnames. Copelands are obvious as he was a Copeland. I don't know how old he is, but some of the females could be Copelands (for example) now married, such as the 2 to Bryants (who could have married Bryant brothers) & 2 Lassetors. (Frankey was common as a nickname for girls named Frances or Francesca though we would consider it more of a boy name now).

I would first figure out what you DO know or can easily figure out - i.e. how old was he when he died, how many times and to who married, and which named children are biological with different surnames due to being grown married women and which could be or are are step-children he may have adopted (formerly or informally). I don't know if it was common practice to name the children chronologically or how / when that might vary, but you might have some clues from other records on that. (My guess is chronologically here at least to a degree because not only are they not all separated by gender, but the 3 who clearly don't seem to belong that you are interested in, are thrown in the middle of the presumably biological children by the wife - i.e. Copeland males and females. But don't quote me on that!).

That being said, as I first skimmed, my first thought was it would probably be more common & acceptable (PC) to recognize in your will your illegitimate ("white") children than your slave-borne children and that he was "owning" up to them. And upon closer read, I still think the same.

Notice those middle 3 are named William Johnston, Sarah McIver? and Richard Johnston and he states "whom I own as my children although Known by the Name of Johnston and McIver." That is in reference to the same 3 children who clearly have last names who vary from his own & as 2 are male, this to me, argues a strong case for illegitimate "white" children (born out of wedlock, but not due to slavery).

For one thing, most slaves would either have no surname or would carry that of their masters.
I don't know if they always carried the master's surname or only took their last master's name once emancipated (or if run away, if even then). I would assume it would be more the norm of course for such cases of slave relations to involve the slave owner and his OWN slave and as he is named Copeland and not Johnston or McIver, I doubt he would be having relations with his neighbor's slaves (though not necessarily impossible), not to mention, I would assume he might prefer one slave vs. several. (There is no polite PC way of putting this, I grant you).

I also think that in such cases where a slave owner acknowledged his slave-borne child, it was still on the QT or DL - - not common knowledge or at least not discussed in polite company. (People may KNOW, but I doubt they'd speak of it openly, even in a will).

That acknowledgement might be protecting them from a distance - but not necessarily naming them in a will or acknowledging them point blank. It might be keeping them as a house slave vs. a field slave, not selling them, or even passing them off to a family member who might know the relationship and protect them but not under the scrutinous, non-approving bitter eye of the wife / lady of the household.

I have a distant Chatham ancestor who apparently had his slave-borne son sent off to live with
his sister's family. It was interesting to learn about - he became a great man - Abraham Galloway - Union spy, abolitionist etc. http://blackamericaweb.com/88762/little-known-black-history-... . (The article mentions the slave was owned by a relative, but if I am not mistaken, my prior research was that she was passed to the relative after she got pregnant or bore his son).

All that being said, you could probably also clarify by other later records, at least on the male Johnston sons. No idea how old they were, but did they own land, were they buried in a white or black cemetery, were they listed in early census records as heads of households (indicating likely not slaves) etc. If he "owned" them and his will did not point-blank emancipate them, it seems he was likely "owning up to" them and not "owning them" as slaves. Some men freed their slaves, even non-biological ones upon death, but you would think they would at least be more inclined to do so with their biological ones upon death if not sooner. Furthermore, usually an owned slave is given (as property / inheritance) to a named biological relative, such as a wife or child and there is none of that indicated in this will. i.e. if he then turned around and gave McIver and Johnstons to a Copeland son, it'd make more sense that he was owning them as slaves and not "owning up to" them as children.

Also, in such cases there are what is called "Bastardy" records that can often be found in the court records. It is usually in cases where a single woman finds herself pregnant and I though I don't know the legal intricate details, I imagine it is much like today when a woman takes the presumptive father to court for child support (even if they weren't married). Often the men admitted it, though I don't know what happens if the man didn't "own up to" the child. (No DNA tests!).
That might further help confirm if you could find such records.

Assuming they are not found in the Bastardy records though, you might just have to look for other such clues. If they lived long enough to be enumerated in 1850, and they stayed in the area or you have approx. death dates, if they were listed as heads of households, they would presumably be white of course, even if it didn't state a race. Or if they were free it might say B for black or M or Mu for mulatto and that might be a good indication.

If they lived to a ripe old age or for some reason you have Johnstons mixed in later with your known Copeland census records after the Civil War, if they were Black or Mulatto, you might infer they were freed slaves who had a special bond or kinship with the family. Often they chose to stay on even where no familial relationship existed if they were treated well, but if they did stay on, this too might indicate a special relationship.

One of my Galloways (from this same bunch from Chatham, but not of the same father of the former slave Abraham but a distant cousin of his) who came to AL then MS married a McCord. His McCord wife died at or shortly after childbirth and so the child's unmarried aunt raised her. (The father remarried a few times & doesn't appear to have been much involved, but who knows). That orphaned female child grew up, married, moved away and a black McCord that I suspect was a former slave of her maternal grandfather's (& aunt's) ended up living with the now-grown adult orphaned child as her servant cook. I don't know there was any hanky-panky between this black McCord & the white McCord (presumably her slave-owner), but she appears to have been unmarried and having children with no known husband in sight and she kept the name & stayed in the family after emancipation. And some of her other known or suspected relatives (children, sister or mother? can't recall exactly) were with other allied Galloway-McCord families as well into 1870 & 1880 MS.

I hope this helps. It's a good question, but I think the clues overwhelmingly IMO point to them as illegitimate (presumably white) children, not children born to a slave that he owned as property.