Pathfinder's magic rules say that a creature with "special resistance to magic" can voluntarily fail a saving throw against a spell.

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 can suppress this quality.

But the rules for spell resistance seem to imply the opposite, that a creature would need to drop its spell resistance before being affected by the spell.

A creature can voluntarily lower its spell resistance. Doing so is a standard action that does not provoke an attack of opportunity. Once a creature lowers its resistance, it remains down until the creature's next turn. At the beginning of the creature's next turn, the creature's spell resistance automatically returns unless the creature intentionally keeps it down (also a standard action that does not provoke an attack of opportunity).

I'm confused how these rules interact. If a creature with SR hasn't lowered their SR, can they voluntarily fail the save without a caster level check from the caster? Or does "special resistance to magic" refer to something other than SR?

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.

A creature with spell resistance can willingly fail its saving throw.

Spell Resistance and Saving Throws are 2 different things. However, in order for it to be able to (not) make a saving throw, it has to be affected first. This means, the spellcaster needs to overcome the spell resistance of the creature in order for it to choose to fail the save, because otherwise the spell doesn't affect it.

Relevant Information

Spell Resistance (Ex)

A creature with spell resistance can avoid the effects of spells and spell-like abilities that directly affect it. To determine if a spell or spell-like ability works against a creature with spell resistance, the caster must make a caster level check (1d20 + caster level). If the result equals or exceeds the creature’s spell resistance, the spell works normally, although the creature is still allowed a saving throw.

Alternatively, it can lower its spell resistance allowing spells to affect it without having to make caster level checks to overcome its spell resistance. It could then choose to fail its save against spells that target it. However, this also leaves it open for hostile spells to affect it as well.

Note: A superstitious barbarian can never voluntarily fails its saves when raging.