This "authority" is in error; the past perfect is not necessarily required here, although there are circumstances in which it would be preferred in formal discourse, and the simple past is not incorrect, although there are circumstances in which its use would be deprecated in formal discourse.
This is in the first instance a matter of context— specifically, what you are trying to express and what time you are talking about, what grammarians call Reference Time.
Perfect constructions do not express events prior to Reference Time; they express a state grounded in prior events which obtains at Reference Time. So if you are talking about a time in the past when your subject's previous unfamiliarity with bears was of immediate relevance, the past perfect will be appropriate:
As she was hiking in the Tongass National Forest she was suddenly confronted by a menacing dark creature her own height and twice her mass. She did not recognize it—she had never seen a bear before she moved to Alaska.
But if your Reference Time, the time you are speaking about, lies before the time at which she moved to Alaska, a past perfect will not be appropriate:
She was in her youth a keen student of wildlife—kangaroos, wallaby, emus—but she never visited zoos, which depressed her, so she never saw a bear before she moved to Alaska.
In less cut-and-dried circumstances, usage varies: formal usage and British colloquial usage tends to employ the past perfect more often than US colloqual usage. If you have an appetite for technical grammar, I modestly recommend our Canonical Post on Perfect Constructions, particularly sections 3.2 and 4. If not, the best 'rule of thumb' respecting use of the past perfect is FumbleFingers‘ Perfect Truism:
”Don’t use the perfect unless you need it.”
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
The first and most important thing you need to understand about using the past perfect was expressed by FumbleFingers here.
The reason your friend found your sentences ‘busy’ is probably that the word before expresses everything you want the past perfect to express.
The second thing to understand, about perfect constructions in general, is that a perfect construction does not narrate a prior event, it expresses a current state which exists as a consequence of the prior event. For instance:
That is, the past perfect expresses the fact that when you arrived you had a shovel in your possession, because earlier you had packed one. In cinematic terms, it's not a ‘flashback’, a jump backward in time, but a sort of temporal ‘pull out’—the ‘camera’ backs up so the field of vision brings the past into view, but it remains as it were focused on the point at which you discovered the sand.
Consequently, the past perfect should be used only when your narrative has defined a specific point in time (the technical term is Reference Time) at which the state established by the prior event becomes relevant. This is not the case with your two sentences: