Learn English – How to use “Welcome aboard” right

I think the weirdness comes from the fact that this is not really a conversation, in the sense that it is not an exchange of information between people who intend to be talking to each other.

In some cultures (including the US, where I am), it is considered friendly to acknowledge someone entering (and leaving) your store with some kind of greeting. Asking after their health or well-being is a formula; practically no one expects you to answer the question in its actual meaning.

This is part of a set of formulas that I think practically every culture has for conversation in one way or another. There are papers written on the subject. In this case, perhaps you could think of it as the speaker's way of letting you know that they have noticed you and are free to talk with you; you can reply with any formula of your own ("fine", "ok", "better than a sharp stick in the eye") to acknowledge that they have spoken; if what you say sounds enough like a formula itself (the first two, not the last), then you will not be expected to continue a conversation.

But if you came into the shop looking for something you wanted help with, the fact that they said something might help you know they were free to listen to what you wanted at that time, in case you would otherwise have been shy about asking.

How are you? is a very general question that invites the person to comment on whatever part of their current state is important. I would say the second definition is more accurate.

However, as Alan pointed out in the comments, this phrase is often used as a generic greeting. In those cases the question is not really seeking information, it's an expression of politeness and consideration for the other person. In those cases, it is polite to reply something like "fine" or "good", followed by "and you?".

It's a kind of complex phrase. Even as a generic question there is some room to provide a somewhat meaningful answer depending on how you actually feel, for example saying "I'm alright" if you're not doing so well, or "Great!" if you really feel good. It's impolite and off putting to respond very negatively or with too much detail, e.g. as in telling a stranger about your personal medical problems.

Moreover, the meaning of the question can become even more vague between people who know each other very well (well enough that a truthful answer would be appropriate). For example, if I see my brother and say "Hey, how are you?" I may or may not actually be interested in a truthful and explicit answer. This generally leaves it up to the person being asked to decide how much information they want to provide.

Other similar phrases can be used to indicate more of a generic greeting or more of an actual question. For example, I would say What's up? is more of a greeting, and usually doesn't even require an answer. On the other side, How are you feeling? is more explicit in seeking an actual answer from the person about how they are feeling.