I think your example eviscerates the first "Increase in Probability" definition, at least as presented here, and shows that it doesn't account for non-independent evidence.

There's a deep philosophical point at stake here. Is probability a 1) quantification of a person's uncertainty, or 2) a statement about the universe?

2) is a position that is not well-regarded here, and I would recommend Probability is in the Mind and then possibly Probability is Subjectively Objective as to why.

If 1), then their previous knowledge matters. It's one particular person's uncertainty. You start off very uncertain about the lottery result. Then you read the first newspaper report, which makes you much less uncertain about the lottery result. Then you read the second newspaper report, which makes you very slightly less uncertain about the lottery result. As you watch your uncertainty decrease, you see that the first report has a huge effect (and thus is strong evidence), and the second report has a small effect (and thus is weak evidence). With more background knowledge about correctness, the second report drops to 0 effect (and thus is not evidence to you).

The mathematical way to think about this is that the strength of evidence is a function of both e and b. This is necessary to ensure that there's a probability shift, and if you don't force a shift, you can have even more silly 'evidence', like sock color being evidence for logical tautologies.

One might object that they're not particularly interested in measuring their personal uncertainty, but in affecting the beliefs of others. If you wanted to convince someone else that Bill Clinton is probably going to win the lottery, it seems reasonable to be indifferent to whether they read the Times or the Post, so long as they read one. But your personal measure of evidence is wildly different between the two papers! How do we reconcile your personal measure of uncertainty, and your desire to communicate effectively?

The answer I would give is being more explicit about the background b. It's part of our function, and so let's acknowledge it. When b is "b & e1", then e2 is not significant evidence. When b is just b, e2 is significant evidence, and so if you want to convince someone else of h and their current knowledge is just b, you can be indifferent between e1 and e2 because P(h|b,e1)=P(h|b,e2).

Let's further demonstrate that with a modification to the counterexample like the one Manfred suggested. Suppose I learn e1, that the New York Times reports that Bill Clinton owns all but one of the tickets, by reading that day's copy of the Times, which I'll call r1.

Suppose my background, which I'll call t, is that the New York Times (and my perception of it) is truthful. So P(e1|t,r1)=1. I liked the prose so much, I give the newspaper a second read, which I'll call r2. Is r2 evidence for e1? Well, P(e1|t,r1,r2)=1, which is the same as P(e1|t,r1), which suggests by the "increase in probability" definition that reading the newspaper a second time is not evidence for what the newspaper says. (Note that if we relax the assumption that my reading comprehension is perfect, then reading something a second time is evidence for what I thought it said, assuming I think the same thing after the second read. If we only relax the assumption that the newspapers are perfectly correct, we don't get a change in evidence.)

Does it seem reasonable that, given perfect reading comprehension, that considering the same piece of evidence twice should only move your uncertainty once? If so, what is the difference between that and the counterexample where one newspaper says "Times" on the front, and the other says "Post"?

(If you relax the assumption that the newspapers are perfectly correct, then the second newspaper is evidence by the "increase in probability" definition, because of the proposition g discussed in the OP.)