The tiger, who died promptly before James died, was a/the representation of James.
Yes, regardless of the rather dubiously-helpful status of 'first mention', the representation does, within the context of this isolated sentence, imply that there is only one representation, while a implies that there may be more representation.
From your comments:
teacher is pointing at a/the written topic on the white board.
I think both are acceptable, with "the" emphasizing that this is the topic she is pointing at, regardless of how many topics there are, and "a" providing the sense that there may be more than one topic on the whiteboard, but is used because there is no intention to emphasize.
Yes, both could be possible. It is not really a matter of emphasis. It is just a matter of signalling to the reader the same information as the articles did in your example with a/the representation. One cannot talk about everything an indefinite or definite noun phrase is doing, or could be doing, in a sentence that has zero context. In a larger context, the topic could be the one under discussion, as opposed to others that are not, In this case, the is doing a job that it frequently/usually does: telling the hearer that he should be able to identify which topic the speaker is referring to.
it is hard to derive a perfect understanding from a/the context.
The emphasizes that context in the sentence is the one you are trying to gain a perfect understanding from, while a is generalizing.
Rather than emphasize I prefer the word signal. The articles usually signal something to the reader. Very often, the in a noun phrase signals to the reader that the reader should be able to identify the referent (which thing the speaker is referring to). The context means the context that I believe/assume that you can identify.
On the other hand, a definite noun phrase such as the context can, in some contexts, signal to the reader that I am going to make the referent the subject of what I am going to talk about next, and after I do, you will be able to identify it:
That's when the tall guy got here.
What tall guy? Which tall guy? The tall guy I am about to tell you about.
The indefinite article does not send the same signal. It is, as you say, just generalizing. A can imply that there may be more than one of something. This is probably what it is doing in all of your examples.
On the other hand, in some contexts, a is used more like one (this is not surprising, because the indefinite article an came from the word one):
I have a bicycle and two cars.
A very often means no more than one here.
Keep reading Abbott. I don't know if you have read her definite and indefinite (pdf). It is one of several that she has written that have similar titles.
I like to write the stuffs down on the book when I annotate a book.
First of all stuff is a mass noun and not used in the plural. If someone said stuffs they were using the word in a rare and unusual manner, perhaps humorously. (The word foodstuff is used as a count noun.)
Here, in both cases the is used to signal to the reader or hearer that he or she should be able to identify which stuff and which book the speaker is talking about. The speaker assumes the hearer can identify which stuff and which book he means. As a matter of fact, I personally cannot do this, and I have no idea which stuff and which book are being talked about. There would have to be some other way–existing outside of the text of this sentence–for me to identify which ones he means.
One of these ways is that the speaker could have mentioned them before. This is where the so-called first mention guideline comes into play (I could never call it a rule because it simply isn't.)
Another way is that he is holding the book in his hand and obviously referring to it as the book. Again, this is not so much an emphasis but a subtle signal: there's no other book in sight, so he must be talking about the one he's holding. In this case he could also say this book.
Likewise, perhaps it is obvious from context which stuff he is talking about. For instance, if this sentence was said during a video and it is clear which book and which stuff the speaker is talking about, then the speaker can use the definite article with them.