I am also trying to become a lucid dreamer (with full awareness during the dream). I succeeded several times, but only very briefly. The excitement causes me to wake up. But the best way for me to know I'm in a dream is when I am reading some text. Texts are inconsistent and when I reread a sentence it might be completely changed.

This is exactly what happened to me just last night. I was able to read a page from a book, which was visually perfect, as if I was reading a real book. However, once I read it and diverted my gaze, then looked back to it, some of the words had changed (not all). This made me realize I was in a dream.

My waking ability to visualize is poor too. I have a very vague mental picture of what my wife's face looks like, and I cannot see it all at once. Nor can I visualize an entire chessboard in my head, despite being able to play chess blindfold well enough to beat novices.

A good question. I've developed the same test that the "rationalists" sometimes use to tell if you have an opinion. If you can remember something in one way, and it seems to fit, but you can remember something in another and it equally seems to fit, then you don't really remember the detail of the two ways it seems to fit, you just have a vague idea.

I can draw you a map of my (real life) office. It will be fairly accurate within what I can see, and I can put down specific names. If you swapped two names, I could figure out that does not corresponding to my memory of the office. I remember when we went out to a team lunch two months ago, and I remember the specific order we sat in the booth, and many details about the specific booth we sat in. And to be honest, I would rate my memory as below average.

By contrast, I call to mind a dream right now in which I encountered a crowd of creatures on a hill. I definitely remember it was a "crowd". But... were there a few of them? Yeah... I suppose that fits. Or were there hundreds of them? Yeah... I suppose that fits. I remember they were about 1 foot high and basically the critter from Aliens... but did they have claws or hand-like things? Either fits my memory. Did they have long tails? I think so, yeah. Were they articulated or solid? Yeah... either way, really. I remember them as a "crowd of creatures like the aliens", but when I really drill down... I don't actually have that many details. Were I an artist, I could draw you a picture that fits my dream fairly well... but I could then turn around and draw a picture that in the waking world and light of day would be fairly substantially different, but that my brain would say matches the dream just as well.

I've read things too... but... what color were they? Usually I don't have an answer to that. (I think the common idea that "people dream in black and white" is inaccurate... people dream in not color. They encounter concepts like "words written on a sign" and they just plain don't have color at all. I occasionally do remember explicit colors, but it's an exception.) What was the font? Was it all uppercase? Yeah... maybe. Was it spread across multiple lines? I dunno, I can remember it either way pretty equally fluidly.

Don't get too caught up on these specific examples; for instance I can call to mind a specific instance where I would say concretely this text was on two lines, specifically. I'm not saying these details never appear. What I am saying is that for most people, I suspect if they examined their own dreams like this they'd discover it has less concrete content than they thought.

I would also suspect that there are the exceptional people out there who do dream highly specific dreams every night. But I bet they're the exception. Hard to know. Given the evidence that the act of recalling memories involves destructively reading them and then rewriting them back out, I would also suspect there's another group of people who actually have vague dreams and turn them concrete in the very act of remembering them, but I won't be able to prove or disprove that hypothesis pretty much until we have full brain simulation, so take that with the appropriate levels of salt.

(This is also useful for real life memories. I chose concrete memories I'm confident about like my office layout. By contrast, something like "the wall color or pattern of the hotel I visited a couple of months ago"... well... I know it's in the "boring beige" spectrum but beyond that I can mentally decorate it in any number of specific ways without it triggering any conflicts with my memories, so, I don't remember that. By contrast, I do remember the layout reasonably well. But I don't remember the counter colors, or even whether they were formica or granite.)

It also does not feel as if I am fabricating the details upon retrieval of memory, since the memory of the sensory information is immediate and fades with time, the same experience as when you see something, but then look away. At first, you'll have a strong memory of the sensory experience, but gradually it fades. So unless I am for some reason able to instantly fabricate strong sensory information from a vague concept only upon waking up, but not, say, hours or days later, then this theory does not explain my particular experience. For some particular dreams, I remember vivid imagery and other sensory information years after they happened.

Meditation and dream journals seem to increase the effect of either having or remembering dreams with vivid sensory information for me.

Your comparison with waking memories and experience seems apt, because the two phenomena (dream experiences and waking experiences) seem like they are one and the same, perhaps only differing in average ability to retain memory or salience of experience. And just as some dreams are less salient than others (at least if going by strength of memory), waking situations tend to also differ in this regard. When you are on autopilot or lost in thought, your awareness of your surroundings and of the happenings around you decrease, and so does the salience and strength of the resultant memory of the experience. This makes me think the only real difference between dream experience and waking experience (for me) is the level of awareness/consciousness, which tends to be lesser for dreams, but not necessarily so (i.e. there is overlap).

Your example of a dream problem scenario was amazing, by the way.