The story you cite can be found here. Rabbi Zilberstein also discussed ניחוש in the second volume of his והערב נא, on לך לך.
The prohibition against divination does not include a logical prediction based on science. So there's no prohibition for someone to decide to carry an umbrella based on the forecast. On the other hand, it would be a problem if he were to buy a heavy coat if a groundhog saw his shadow on February 2!
Furthermore, it's not considered divination if the person doesn't rely on the omen entirely. Eliezer didn't rely solely on the signs when he gave the gifts to Rivka. He had first inquired about her family. See here for a brief treatment.
Rabbi Zilberstein in the above cited והערב נא discusses a case of someone who was planning a trip. The money set aside for the plain ticket was kept in a drawer. The night before, a fire broke out in the apartment- and was extinguished. There was no damage, except to the very drawer in which the money was kept in. The individual inquired whether he should take this as a sign not to travel, or whether such concerns would be considered divination. Rabbi Zilberstein writes that in case of "דבר של פלא"- a wondrous occurrence, one can be concerned and not violate the prohibition of ניחוש. (This case can be read here.)
Now spilling a cup is not an unusual event. (In fact, someone just spilled a cup as I was typing!) But here, the event happened at the last possible moment. And it wasn't an event that was unconnected to the matter of the divorce (consider, in contrast, if a bird would have flown in through a window at that moment). The spilling had a direct effect on the divorce. At the very least, it would have delayed it by requiring the writing of a new גט. Arguably, one could say this was not some common omen, but a highly rare happening- a פלא.
I will also make an assumption about this couple's circumstances. I doubt that this was a case where remaining married would have been madness- i.e. where there was abuse, etc. In a case like that, it's unlikely that the both parties would have paid attention to the omen and spend the rest of life in misery. That both the husband and wife were willing to stay married indicates that the marriage was worth salvaging. When they decided not to proceed with the divorce, they implicitly decided to spend more effort at repairing their relationship. (Of course, this is somewhat speculative, but I believe it to be very reasonable.) Thus, they didn't rely entirely on the sign when making their decision.
Thus, this case had two aspects that would avoid the prohibition: it was a very rare bizarre occurrence, and the couple relied on other factors when they made their decision. Both these aspects are absent in the case of someone dropping a walking stick. (By the way, is it correct to translate מקל specifically as "walking stick" as oppose to a "staff" in general?)