As already stated Haldane's rule doesn't always apply to odds of sex being born, it usually applies to the odds of that hybrid being fertile based off of whatever sex it actually was. Haldane's rule can also apply to the odds of a sex being born at all, though this situation is run into less often. So in short, no the vast vast majority of the answer to your first question is no, it doesn't mean that only one sex is produced. The majority of the time when Halden's law applies it only affects fertility of the offspring not the production of them.
Let me get into more specifics though, to help clear things up better. I think I detect two main conceptual mistakes which may be leading to your other confusion, so let me address them and see if it answers all your
confusion. At least one, maybe both, seem related to the way you quoted Haldane's law, so let me first quote a slightly more accurate version of the law, straight form wikipedia:
When in the F1 offspring of two different animal races one sex is absent, rare, or sterile, that sex is the heterozygous sex (heterogametic sex)
Halden's rule doesn't apply to all hybrids
The law starts by saying "when in f1 offspring one of 3 situations occur it will be the heterozygous sex. I think the first confusion is simply an attempt to apply this rule a little too broadly instead of limiting it to these categories. In your example of the Pheasant's it appears that both sexes of hybrids are fertile. Since neither sex is rare or sterile Haldane's rule simply doesn't apply, the precondition isn't met so you can't say anything about they heterozygnous sex.
In the case of the doves, well I'm not an expert at this but as far as I can tell Barbary and ringneck are two names for the same dove so I'm not sure that is a hybrid at all. I may just be misunderstanding dove nomenclature/classifications here, but in any case it looks like most doves hybrids are fertile so if I had to guess Haldane's rule doesn't apply in this case either.
However, just because your specific examples may not apply that doesn't necessarily invalidate the entire question. It is certainly possible species where Halden's rule applies where children of both sexes are born, in fact this is usually the case. Thus let's get to the other confusion.
Haldane's rule applies when one sex meets listed conditions
One confusion you seem to have is that the rule seems to imply that fertile hybrids exist despite the version of the rule you quoted specifically saying that halden's rule applies when hybrids are sterile. That's because the actual rule applies only to one sex, meaning that one sex will be infertile and the other sex fertile. If both sexes are infertile, as with the mule, Haldne's rule doesn't actually apply since it's not a trait unique to one sex(except when it does...I'll come back to that).
When precondition X applies the heterozygnous species demonstrates that condition
The law states that if offspring are "absent, rare, or sterile" that sex is the heterozygnous one. Those are three conditions any of which can apply, this doesn't mean all the conditions apply. The most common examples of Halden's law your likely to run into is where one sex is sterile, in which case that sex is sterile, that doesn't mean it has to be sterile and rare and absent. For that matter in cases where one sex is rare it's possible that the sex will be fertile on the odd example it's born despite being rare.
So back to what I said at first, most often Halden's law means two sexes that are equally common, but where one sex is sterile when it's born. The best example that I can think of right now is with bovine hybrids, where cattle were often crossed with other bovines to get the 'best' bovine to raise for meat was common. Crossing bison with Cattle produced more resiliant and better animals that were easier to raise, the only problem was that males were born infertile while females were fertile. This resulted in an inability to keep hybrid breeding populations without males to mate and thus keeping a 'feral' breeding pool of bison cows to mate with cattle bulls; which was a problem since bison weren't domesticated and keeping breeding stock was an issue. Thus a very common hybrid ran into issues due to Haldane's rule, keep this issue in mind, I'm coming back to this example later for my last point.
It is true that some hybrids have only one sex common due to Haldne's rule, I can't think of an example right now but there are many, just none exist in the common big/domesticated mammals I'm most aware of. Usually if a sex is rare that sex is likely to also be infertile, the issues that cause the sex to be rare will usually also make it infertile. However, as I understand the genetics it's entirely possible for a species to have one sex be rare, but that sex be fertile when it is produced; in fact id love an example if anyone can suggest one!
Laws are really just rule's of thumb
You don't want to apply this, or really any genetics law, too exactly or biology is going to decide to throw you a curve ball. Because of the oddities of the ways genes are combined, and even the possibility of random mutation, many 'impossible' situations still come up. Look at mules, thought to be infertile and yet a half dozen fertile mules have been documented at least, in fact most common 'sterile' hybrids I know of have examples of fertile hybrids being produced.
Which brings us back to the beefalo example above. Remember keeping bison cows for breeding stock was a major problem for them? Well that problem solved itself when a fertile male beefalo was born. Even though they are almost always infertile this male was fertile, as were both sexes of his offspring. This was due to random chance (I don't know if it was mutation or just the combination of the genes specifically), but a rule was broken by genetics. Thanks to this we were able to start up hybrid breeding populations of beefalo, all decedents of this male, beefalo are the most common bovines raised for meet now of days!
The point being go in expecting any rules to be bent in genetics at time. There are exceptions to most of these rules, life finds a way yadda yadda.
Hope all that rambling gave you a better idea of how the rule applies.