What is difference between "on suspicion of conspiracy to commit murder" and ...?

Conspiracy to commit murder is using murder as a verb, conspiracy to commit the murder is using it as a noun. No difference in meaning at all.

commit = verb
commit murder = verb + verb according to your opinion.
It does not hold water. It is a weird grammar.

Murder is a noun in both cases.

The difference is that in the first sentence, murder refers to the generic crime of murder. He is committing a crime. The crime is murder.

In the second sentence, "the murder" refers to a specific act of murder. He committed this specific crime. He committed the murder.

The phrase "conspiracy to commit murder" is rather common.

By the way, you can make this distinction with many nouns.

He is eating fruit (a generic statement on the type of food being eaten).
He is eating the fruit (referring to a specific piece or plate of fruit).

He wears pants.
He wears the pants of the family.

All three were detained on suspicion of the (redundant) conspiracy to commit the murder of Lee Rigby. It's fine but it needs more information than the first sentence.
"Conspiracy to commit murder" is the wording used in many jurisdictions. The "exact wording" or legal definition of the crime committed.
The law is generic but succinct. You get charged with a speeding ticket, not driving a porsche at Mach 10 - even though that is what you were doing.
If you've killed multiple times, you will be charged with four (etc) counts of homicide, not the homicides - unless you plan to fill in more details.
Suspicion to commit murder could mean you planned it or attempted to kill. If you hired a hit man, you could be tried with the above offense, whether or not you succeeded.

What does collo mean?

I think it is Ori's home-made abbreviation for "colloquial".

Collo refers to colloquial. I got it from Mr. Max

I just followed suit. So it is a mistake I mistakenly sadly learned.
Get started to forget it now.

Without 'the' is both formal and collo, Ori.

We don't use 'the' because the description of the crime is not specific to one incident. It covers all potential crimes of this nature.

But the conspiracy by the three is specific, JTT.

There is no conspiracy at this time. There is the suspicion of same. They were detained on suspicion of conspiracy to commit murder.

I believe I can say that laws/statutes are written in a generic and formal fashion.

People are charged with breaking a law that remains general to all, that anyone could be charged with. The specifics come later.

Here are more examples. I doubt that you'll find any example using 'the' when it describes an actual law/statute

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Tristan Brian Arthur Stahley has been charged with one count each of first-degree murder, third-degree murder, and criminal possession of a weapon after stabbing his girlfriend, 17-year-old Julianne Siller, to death.

Read more: http://www.dailymail.co.uk/news/article-2331440/Pennsylvania-teen-charged-murder-stabbing-17-year-old-girlfriend-death-hiking-trail-town-park.html#ixzz2USnt2Vuq

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He is charged with first and second-degree murder, sexual assault and burglary in the deaths of Touch Chea, 71, and his wife, Sorn Sreap, 73.

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Two men have been charged with endangering an aircraft after a Pakistan International Airlines passenger jet was diverted mid-flight over London