Mark Liberman writes an interesting article on the history of this spurious "rule" in his Language Log blog.
There are some usage guides and older grammars (and the distinction between the two is important) which decree that an adverb shall never intervene between the parts of a compound verb. This so-called "split verb rule" is still being taught here and there, and perhaps due to the continued use of older textbooks in some non-English speaking countries, may be more often encountered by a new learner of English outside of English speaking countries than by a student in the U.S., the U.K., etc.
The current writer, having been taught the language in the mid-1950's in the U.S., was inculcated with this nonsense. However, the truth is that there is nothing in English grammar which prohibits separating the parts of a compound verb.
Where this "rule" arose is not known, but the Fowler Brothers, Henry and Francis, are prime suspects, because their The King's English (1908) was taken in its time as "gospel" by many teachers of the language (and is still viewed as authoritative by some.)
Today, as in 1908, there is no grammatical rule that requires that the parts of a compound verb like "would have liked" be forced together when good sense dictates that they should not be. You can and should write any of:
- I would always have liked to have...
- I would have always liked to have...
- I always would have liked to have...
...or even:
- I would have liked always to have...
If you should encounter an instructor who insists that the parts of a compound verb must always be yoked together, refer that person to the Language Log link above (or send them here!)
The difference is that Past Perfect serves a different purpose in the first part of the sentence (I saw him / I had seen him) than in the second (before he saw / had seen me). In the first part, it's used to establish a timeline - the "past in the past" use that most English learners are primarily familiar with - while in the second, it's used to establish the situation as unreal (in a manner similar to the third conditional - cf. "If I had seen him, I would've said hi").
So going through your sentences:
[11] "I saw him before he saw me" and [12] "I had seen him before he saw me" - You did see him, and he did see you. No unreal/hypothetical situation here. Past Perfect is optional, since you don't need it to clarify the timeline - you're already using "before". You can still use it to emphasize that it's important you saw him first, though.
[13] "I saw him before he had seen me" - You did see him, and you stipulate that he would've seen you, but he didn't (eg. because you managed to sneak out first). It's analogous to [11], but since you want to signify that he didn't see you, you dial the Simple Past back to Past Perfect.
[14] "I had seen him before he had seen me" - analogous to [13], but I'd say that in this case you're placing the entire situation in the context of a "past before the past". Eg.
I saw John a week ago, and he asked me if I was at the birthday party. I was, but I had seen him before he had seen me, so I snuck off.
As for the other examples:
Sally stopped Ted before he had a chance to reply.
You don't need Past Perfect here ("had had"), since there's no ambiguity - "stopped" already implies taking away the chance to reply. In a different sentence (eg. "Sally smirked before he had / had had a chance to reply") you might have to use Past Perfect to indicate that one action prevented the other from happening.
She left the country before she wrote / had written her thesis
I wouldn't go as far as to say Past Perfect here implies she began writing. It's really more about Simple Past here meaning that the entire process of writing the thesis to completion happened after she left the country, while Past Perfect means that - at the very least - she hasn't completed writing, and she would have if it wasn't for her leaving the country. It might mean she started writing, or that she was strongly planning to start.
Snape had struck before Harry was ready, before he had even begun to summon any force of resistance.
A literal reading implies that Harry was ready after being struck, but has never begun to summon any force of resistance - but in this case I'd say it's more about gradually putting more emphasis on how much of an interruption Snape striking Harry was.
Best Answer
Some people might think OP's original text using the Perfect Infinitive (to have seen) is more "logical" because it more explicitly echoes the past tense element of would have liked. But note this from Garner's Modern American Usage (2009)...
Having said that, Garner himself acknowledges that the erroneous phrasings are very common. In fact, Google Books claims 386,000 written instances of would have liked to have seen. That's not so common as the "correct" version would have liked to see with 1,050,000 instances, but I think it's enough to justify saying that unless you need to pass a "fussy" exam, you could reasonably call it a stylistic choice.
OP's "Simple Past" version ✳I would have liked to saw is idiomatically and grammatically unacceptable.
EDIT: I see the question has been edited to tell us that the supposedly "incorrect" (according to Garner) version actually comes from a BBC Learning English page. I take this as further evidence (if it were needed) that Garner's position is unjustifiably pedantic/prescriptive. And comparing American and British charts on Google NGrams, I see no evidence that his position reflects any kind of AmE/BrE usage split.
At the risk of stoking controversy on what I consider to be something of a non-issue, I'll just cite this from grammarphobia, who also seem to have little time for Garner's position...