Monk's Vinaya is not SīlabbataUpādāna

What comes with a set relates to the other members. Sīlabbata can't come alone, so it's meaning already came in its context.
Depentent origination means aggregates' relation.
Taṇhā means craving, addict.
Upādāna means caving again and again.
Diṭṭhi means the mental habit of the misunderstanding in the aggregates' relation.
DiṭṭhiUpādāna means caving again and again of the misunderstanding in the aggregates' relation. This is only unwholesome.
Sīlabbata means doing verbal and physical habit, depending on Diṭṭhi. This is able to arise with wholesome, unwholesome, and neither wholesome nor unwholesome, e.g. the thinking "I want to be a Deva in heaven" is arising with SakkāyaDiṭṭhi (unwholesome) but next moment the doing giving a gift, follow that SakkāyaDiṭṭhi for going to heaven, is arising with wholesome.
SīlabbataUpādāna means caving again and again of doing verbal and physical habit, depending on Diṭṭhi.
See DN MahāNidānaSutta:

So it is, Ānanda, that feeling is a cause of craving. Craving is a cause of seeking. Seeking is a cause of gaining material possessions. Gaining material possessions is a cause of assessing. Assessing is a cause of desire and lust. Desire and lust is a cause of attachment. Attachment is a cause of possessiveness. Possessiveness is a cause of stinginess. Stinginess is a cause of safeguarding.
Owing to safeguarding, many bad, unskillful things come to be: taking up the rod and the sword, quarrels, arguments, and fights, accusations, divisive speech, and lies.

The Answers

What does this fetter really refer to?

It refers to SīlabbataUpādāna.

Is it clinging to rites & rituals and thinking that these practises by their own can lead to liberation OR does it refer to clinging to precepts?

It addicts to do follow to DiṭṭhiUpādāna again and again. So if the rites, rituals, practises, and precepts depending on DiṭṭhiUpādāna, it is SīlabbataUpādāna.

So if most monks just follow rules and precepts because the Buddha said so (or they are deemed to be effective), then this is called Silabbata Parāmāsa?

The mong who keep on 91,805,036,000 vinaya rules follow to Buddha because they trust the Buddha, who rightly understanding in the aggregates' relation, is not SīlabbataUpādāna. It is similar to when you are following an expert guide and you are going to arrive the right destination surely, no one can't say "you are going the wrong way" although you do not understand the map by yourself at first.
91,805,036,000 vinaya rules are keeping Tipitaka go on through 5,000 years, but some lazy buddhist people try to cut them off, while there are many monks still can do follow to all those rules. It is superficial thinking and killing Buddhism age because Theravāda Tipitaka still available nowadays by the Tipitaka Memorizer who did follow all those vinaya rules. And it is going like this until 5,000 years. The similitude is the chemist who has recited, memorized, and learned about the periodic table and the formulas from the past chemist generation to generation.
There were many Nikāya tried to cut some part of Tipitaka off, and now their canons already disappeared, right?

If it is the former, then most "rational" inclined people should have little to none of this fetter, no?
If it's the latter, then it's NOT about blindly believing a precept, but questioning it and seeing for oneself that it's helpful.

All ordinary people have SīlabbataUpādāna because they have DiṭṭhiUpādāna. They still have the mental habit of the misunderstanding in the aggregates' relation, DiṭṭhiUpādāna, so they still act follow to that habit, SīlabbataUpādāna.

However, it appears to me that many rules in the vinyana are there to avoid social faux pax and unnecessary conflicts. If we take the not-eating-after-noon precept, we will see that the Buddha has reason for devising such precept; but do those reasons still hold true nowadays? I doubt it.

It may be not important for 21th century, but it may be important for 22th century. The Buddhist age is for 5,000 years, not only for 100 years. Tipitaka should lose after Buddha's death for 1-200 years without 91,805,036,000 Vinaya rules, but it still going on because of the Vinaya rules. Who can understand entire chemist canons without the chemist expert, right?
By my experience with Burmese monks (stickest on rules), Thai monks (sticker), and the westerner monks (not stick), the Burmese monks can understand the whole Pali Canon deeply, while the Thai and westerner monks often distort the Pali Canon by their own opinion then say "Pali Canon is conflicting with each other". For example, the fabrication of MN 44 is talking about Jhāna, and SN 12.2 is talking about Vipassanā by each Sutta's context. But the Thai and the westerner monks mix them together then say "they are conflicting with each other." It is like that because many, almost all, Thai and the westerner monks are lazy to recite and memorize Tipitaka follow to Vinaya rules "the monk can't live without Tipitaka Memorizer Teacher if he has not enough Tipitaka memory, and it is counted as a defense each date, 10 dates 10 offenses."
I understand Tipitaka without cutting any part of Tipitaka off because I always support the monks who doing follow entire Vinaya rules.
How can one understand the action and its effects when one never do follow them all?