Native speaker here – American English (New England, Boston area).
I may be grasping at phantoms here, but I have a very faint impression that when one uses "would always", it implies "on those occasions on which that person would do this thing, they always elected to do it in the following way", while, in contrast, "was always" implies that the person (or thing) so described was multiplying occasions to do the thing.
That is, using your examples,
She would always send me strange birthday gifts.
Suggests to me that on the occasion of your birthday, she could be relied upon to choose gifts that were strange, while
She was always sending me strange birthday gifts.
would suggest that she wasn't waiting for your birthday to send those strange dirthday gifts!
Likewise,
Sam and Mary would always choose the most exotic vacation destinations.
suggests that Sam and Mary's vacation destination choices were consistently exotic, while
Sam and Mary were always choosing the most exotic vacation destinations.
suggests that Sam and Mary have a very, very busy vacation calendar.
This
Ned would always show up at our house without calling first.
suggests that it was Ned's custom to drop by without calling first, while
Ned was always showing up at our house without calling first.
suggests Ned is making a serious nuisance of himself by imposing on us so very frequently. (Note that in this example, the "was always" formulation has a much strong negative valence than the comparative "would always.)
The difference, I think, is that used this was "was always" is hyperbole, while "would always" is merely a generalization.
Your description of "was always" + verb as implying a negative is correct, but it doesn't explain how that works. It is using hyperbole, which is a figure of speech which conveys valence by exaggeration. When someone says of someone "he was always doing that thing!" it is not usually meant literally; it is meant as exaggeration for effect.
In contrast, "would always" may well be meant literally, or close to literally. When somebody says "he would always do that thing!" they may well mean "whenever he did a thing, it would be that thing!"
By way of illustration: I had a friend who would always be late to parties. Famously so. Like, numerous times he arrived at parties when the last of the rest of the guests were saying good-bye to the hosts. Once, story has it, he showed up the day after the party. Yet I wouldn't say of him that he was always being late to parties because he didn't go to parties all that often. It wasn't like his life was full of party-going, only late. But, by gum, when he went to parties, he would be late; you could bank on it. So my saying of him that he would always be late to parties was perfectly, incontestably, literally true.
I thought that site explained it pretty well, but I'll rephrase it just in case:..
We use prefer to say we like one thing or activity more than another. We can use a prepositional phrase with to when we compare two things or actions:
I prefer tea to coffee.
We prefer going by ferry to flying.
We use would prefer or ’d prefer, followed by a to-infinitive or a noun, to talk about present and future preferences:
I’d prefer to go by myself.
Would you prefer a quieter restaurant?
She’d prefer not to drive at night.
So, 'I prefer' means 'I like x more than y.' The comparison can be implied to include all possible options in that category by omitting 'more than y,' as in your example:
She prefers to drink tea.
This means that, as a general guideline, she would rather drink tea than any other beverage. Context can also imply a more specific comparison:
Jim: Have you been to Coffee Unlimited? They have really great coffee.
Sue: No, I prefer tea.
Given the context of coffee, this means that she, as a general guideline, prefers tea over coffee.
We use 'would prefer' to specify a specific occasion:
She would prefer to drink tea.
This means that, in this particular instance, she would rather drink tea than a different beverage. This does not imply that she always prefers tea, just that she would prefer tea in this particular instance.
Here's an example conversation:
Tom: Let's go out to Paddy's Pub tomorrow.
Andrew: We should invite Beatrice too.
Simon: Doesn't she prefer wine? Paddy's wine selection is very overpriced.
Andrew: Yeah, but she likes chocolate stouts almost as much. She would probably prefer to get chocolate stout rather than overpay for bad wine.
As stated, Beatrice in general prefers wine, but specific to this scenario she would rather have an chocolate stout.
Best Answer
Would in "Would you believe it?" is a modal verb. I'll try to explain the meaning using this example:
See? It's a rhetorical question. The speaker is implying that the listener would not believe it if he were told it.
Thus, would is a modal verb similar to the would used here:
It's only that with "Would you believe it?!" you are not really asking, you are saying that "no normal person would believe that" or that "every normal person would be suprised if it were true".
They are somewhat similar, but not precisely. Imagine you are alone in your flat. You walk into the bathroom and see that your washing machine is leaking water on the floor. You will be more likely to say
Instead of
.. because there is nobody around to address this rhetorical question to.