[RPG] Does a demon with the Magic Resistance trait have advantage on saving throws against the Summon Greater Demon spell

It is hostile, but that will not affect charm monster until you lose control

My other answer makes the assumption that hostile means the same thing as fighting against you, but this isn't necessarily true. Taking from this answer to a different question about what hostile means, the DMG (pg. 244) describes the following:

• A friendly creature wants to help the adventurers and wishes for them to succeed.
• An indifferent creature might help or hinder the party, depending on what the creature sees as most beneficial.
• A hostile creature opposes the adventurers and their goals but doesn't necessarily attack them on sight.

Since a hostile creature "doesn't necessarily attack them on sight", this means that a demon doesn't have to be fighting the party to be considered hostile, and clearly demons are hostile to everything with regards to opposing everyone's goals.

Despite being hostile, the demon isn't fighting against you initially because the summon greater demon spells allows you to maintain control over the demon until it breaks free, after which point it attacks any non-demon, including you, the caster.

Whilst it isn't free of your control, it will not fight you unless you fight it first, so it will not give the demon advantage against charm monster for that reason, since that spell specifically says "fighting".

Charm monster (Xanathar's Guide to Everything, pg. 151) says:

A creature you can see must make a Wisdom save, and has advantage if you or your companions are fighting it. If it fails, it's charmed by you until the spell ends or until you or your companions harm it. The charmed creature is friendly to you. When the spell ends, the creature knows it was charmed by you.

Also note that, as described in this question, some demons have Magic Resistance, meaning those demons will have advantage against charm monster anyway, regardless of whether you're fighting it or not, so although it's hostile, and you may or may not be fighting it, it'll still have advantage on the saving throw.

Also note that having used its true name as part of the summoning spell only affects the summoning spell, so it'll always have advantage against charm monster, being a different spell.

A "special resistance to magic" referred to abilities other than spell resistance

In the D&D 3.5e Player's Handbook the description of this mechanic helpfully includes an example of a special resistance to magic:

Voluntarily Giving up a Saving Throw: A creature can voluntarily forego a saving throw and willingly accept a spell's result. Even a character with a special resistance to magic (for example, an elf's resistance to sleep effects) can suppress this quality.

This text didn't survive into the SRD (such examples were widely removed to incentivise acquiring the actual PHB), so it didn't make it directly into Pathfinder, and Paizo didn't add any other examples in its stead. However, it seems that in the original source, it was meant to be the case that a spell's target could voluntarily be affected by things they would normally be immune to - such as an elf suppressing their immunity to magical sleep in order to be affected by a spell - and it does not generally refer to spell resistance, which has separate rules for voluntarily suppressing.

Neither 3.5e nor Pathfinder's rules and published books seem to mention this particular mechanic ever again, so there's no other examples or explanation to draw upon that could clarify further.

In any event, spell resistance and saving throws are separate things; if a spell does not overcome a creature's spell resistance, they aren't subject to a saving throw in the first place, so they can't voluntarily fail it. A creature would have to both actively lower their spell resistance and passively deliberately fail the resulting saving throw in order to guarantee being affected by such a spell.