Which Specific rule overrides which General Rule?
The general rules get overridden or have exceptions made throughout the game by specific rules, which is clearly stated in the PHB under "Specific Beats General"
Many racial traits, class features, spells, magic items, monster
abilities, and other game elements break the general rules...Magic
accounts for most of the exceptions to the rules. (PHB p. 7).
The ring works as advertised for the wearer
What you are dealing with here is two specific rules in conflict:
one of which is the magic item specific rule, and the other is
either the Twin specific rule, or the upcast specific. That means
that an answer needs to rely on a bit of judgment to form a ruling.
Negating a magic item takes an antimagic field, which is an 8th
level spell (magical effect).
Spells create magical effects.
Modifications to spells, such as twin or upcasting, are modifications to a discrete magical effect.
A spell is a discrete magical effect (see What is a spell? PHB, p. 201).
Ruling: the ring wearer gets the benefit of the ring, the other target does not.
The ring's magical protective features is applying its magical effect to a single target; there is a single magical effect affecting the target of the hold person spell, and that magical effect is not an area of effect. The target is the ring wearer. The other target of the spell gets no such protection, because that target is not wearing the ring. (And if it were, it too would get the protection).
This question is more clearly answered from the perspective of the target(s) not from the perspective of the spell caster.
Because you are dealing with multiple "specifics" to deal with Hold Person's "general." The target not wearing the ring gets no protection.
What the caster is doing, by applying an additional resource (be it meta magic points or added spell slots) to the single target spell as cast, either by twinning or by upcasting to choose an additional target, is creating a second instance of a discrete spell effect. The upcast is not creating an area of effect spell, nor is the twin. The only caveat regarding the ring of spell turning not applying advantage to the spell save is the application of an area of effect spell.
At Higher Levels. When you cast this spell using a spell slot of 3rd
level or higher, you can target one additional humanoid for each slot
level above 2nd. The humanoids must be within 30 feet of each other
when you target them. (Basic Rules, p. 93)
Notice that "at higher levels" does not hold all creatures in an area; what it allows is for more than one creature In Range to be forced to save of be held.
The ring's core protective feature is:
you have advantage on saving throws against any spell that targets only you (not in an area of effect.)
Twinned is the easier case to address
The twinned hold person can only target you; it takes the magical effect (and expended resource) of the twinning to apply the discrete magical effect of hold person to another target. From the frame of reference of the ring, a single target spell is being attempted against the wearer. The other target gets no protection.
Upcasting is the trickier case
Because of "how spells at higher level" work, there is some question as to which exception takes precedence: the addition of another target, or the protective effect of a legendary item? The item wins.
- Compare what level of time, effort, and magic it takes to create a
ring of spell turning (DMG. p. 129) 500,000 GP, 17th level minimum,
and (500,000 GP) divided by (25 GP) character days (20,000~ 54.8
years) to create. Note that it takes 8th level spell casting (first
available at 15th level) to get access to an anti magic field, which
would suppress the effects of this magic item.
Why? The second (or third, etc) instance of hold person that comes from up casting does not convert the spell to an area of effect. What it does is create another magical effect that another creature has to save against. From the frame of reference of the ring, that other target may as well not exist. Its magic is only applicable to the wearer. The wearer gets advantage on the save.
To rule otherwise is to apply 8th level magic to lower powered magical effects
In order to nullify the effects of a legendary magical item generally requires subjecting them to an anti magic field: an 8th level spell effect. That's is considerably more powerful magic than a 2d level spell, or the twinning meta magic.
But let's bury ourselves in edge cases, as rules lawyers like to do
Were the Hold Person Spell to be cast at 8th level, the ring of spell turning cannot turn the magic back upon the wearer. That's a limitation of the item.
The spell expands to fill the slot it is put into (p.201)
The instance of hold person that the wearer is saving against is still based only on that person and they make their own save. The other creatures are on their own.
One could argue that having exceeded the spell level threshold of one protective feature may as well override all ring protective features; the hold person effect at 8th level of casting overpowers the ring and no advantage would accrue to the saving throw. It's a reach, but since spells grow to fill the power of the slot when upcast, the case could be made and 8th level is the same level of power/magic as the antimagic field mentioned previously. It's a that goes into the grey area where the rules are not specific, but you can as well argue that the 7th level threshold only applies to the turn, and otherwise the ring fundamentally does its usual thing: protect the wearer from spells (discrete magical effects) directed at them.
A similar case of multiple instances of one spell: Eldritch Blast
Jeremy Crawford has ruled that if multiple blasts of Eldritch Blast hit the same creature, with the repelling blast invocation being active, each instance of the blast moves the creature the 10 feet. This ruling and the above reasoning is consistent with the point that a spell cast with multiple instances (the eldritch blast must hit each in each time for this to work) has each instance treated separately. The caster in this case can spread the blasts to multiple creatures, or to one creature, but each instance has its own discrete magical effect.
That's what a spell does: create a discrete magical effect.
The basic characteristic of the spell, hold person, is that it targets a single creature. The upcast creates a parallel instance that does not effect the other targets. Applying additional instances of a spell by either twinning or up casting a "single target" to add additional targets does not change the nature of the spell.