What do the words of the Mahayana Buddhist mean?

First let's take a simple example. Here you are, reading something I wrote in the internet. But what do you really know about me? Am I male or female? How old am I? What ethnicity am I? What are my attitudes about Buddhism, or politics, or pizza? Doubtless you'll look at my name and make some assumptions, but what makes you think that's my real name? Just because I chose a screen name that looks like a real name doesn't mean it's my real name...

The internet is an interesting test-case because it's such a restricted mode of communication. You can't see me; you can't hear me; you can't get any of the subtle or obvious non-verbal cues you'd have if we were talking in person. You are building an image of me in your head as you read — based in part on what you read, perhaps (though by no means necessarily) — and you will respond to what I write as though it were written by the persona you have lodged inside your head. To that extent, the person you are communicating with (aka 'me') is completely imaginary; it is a set of presumptions and assumptions drawn from your expectations of what a person who says the kinds of things you are reading would be like.

This isn't qualitatively different from an in-person conversation. The only different with face-to-face communication is that you have more sensory data on which to base the assumptions you use to create that inner persona. Possibly my posture would lead you to assume I have certain characteristics, or the way I gesture with my hands, or the intonation of my voice... With an in-person discussion you might paint a richer persona of me inside your head, but it wouldn't be any less of an imagined persona.

But there's one obstacle you will consistently run into that keeps me from being entirely imaginary. Every now and then you will see me do or say something utterly inconsistent with the presumptions and assumptions of the persona you've assigned to me. For instance, I might type the word 'f**k', which may shake up some of your expectations of what a poster on a Buddhist site would say. Mentally-crafted personas of this type are boilerplate: stereotypical, standardized, fixed, and thoroughly un-alive mental objects. Actual living beings never fit well into the mental personas we impose on them; actual living beings are subtle and complicated; they don't sit nicely in boxes.

Now, we could walk through life holding to the principle that people are exactly as we imagine them to be. In fact, most people do exactly that; that is the essence of karmic existence. People hold that others are exactly as they imagine them to be, no matter how shallow or thin that internalized persona might be. People get angry, frustrated, or frightened when others violate the mental personas assigned to them; people make friends, fall in love, or show loyalty when others conform to the mental personas assigned to the; people generally like the world (and the people in it) to match the mental expectations they have developed. They are uncomfortable and discontent when the world or the people in it fail to comply.

However, Buddhism aims to help us release those expectations: both expectations of what ought to happen in the world and expectations of what other people ought to be like. Buddhist practice teaches us to slide past those expectations and see the world (and the people in it) exactly as is, without imposing the constraints of mental objects. Before we start the path, you are talking to a mental construct of me that lies in your head and I am talking to a mental construct of you that lies in my head; We will both occasionally be surprised and discomfited when one of us does something that doesn't fit with the other's preconceptions. The farther we go along the path, the less we rely on those mental personas, and the more we speak directly to the other without mental filters. You exist and I exist, and the problem we both have is getting through our imaginary mental constructs to actually see each other.

No. Imaginary things are fancies generated by the mind. Your eyes and ears do not "imagine." Your mind does. Your eye-consciousness does not imagine visions, but you can use your mind-consciousness to imagine whatever.

When someone interacts with another person, they are not interacting with something that they made up in the manner of a daydream. They are interacting, just like the person quoted in the OP said, with impressions generated via sensory contact. There is no reason to determine from the quoted material that anyone thinks that anyone is something imaginary thought up in the manner of a daydream.

Let's say there's a person called George, who is a lay single unmarried man, aged 39, and is strongly interested in Buddhism.

Now let's take a look at other people around him.

Let's say his friend Tom, is of the same age and is happily married with 3 kids. He looks at George and pities him for being single and not having his own family. He thinks about advising George to try more diligently to achieve the goal of marital bliss in the near future.

Another friend of his, Bob, is an openly gay man. He looks at George and thinks that perhaps George is a closeted gay man and that's the reason why he's single and unmarried. He pities George for not being true to himself and opening up to his fullest potential of happiness.

And then there's his colleague, Jim, who went through a bitter divorce, separating from his ex-wife who cheated on him with another man. Jim is very emotionally hurt from his divorce. He looks at George and envies him for having made the right choice to remain single and unmarried.

Lastly, George knows a Buddhist monk by the name of Sadhu Bhikkhu whom he met at a Dhamma talk. Sadhu Bhikkhu looks at George and thinks that perhaps George is single and unmarried because he is considering joining the monastic order in future. Sadhu Bhikkhu is hopeful that more interested men and women would join the monastic order.

So, in each case, Tom, Bob, Jim and Sadhu Bhikkhu, all know the real person George, but they each form a different mental model or mental image of George. When they interact with George, they are interacting based on that mental model that they have created, and not based on the real situation of George.

I believe this is what is meant by the Mahayana Buddhist.