[Num][Counter]の[N] tends to be used when the speaker is referring to specific instances and the listener already knows their existence, and probably their quantity, too. その２匹の犬は〜 is a typical example of this. When you say “those two dogs”, the count “two” is probably not new information.
３本の花を見た is possible, but it sounds a bit dramatic as if those three flowers will play an important role in the story the speaker is going to tell. This effect is similar to that of titles like 七人の侍, 三匹の子豚, etc. You may not know about those warriors or pigs before you see the movie or read the book, but you can tell those titles refer to specific warriors or pigs and wouldn't question the use of a definite article in The Seven Samurai or The Three Little Pigs. (The official English title for 七人の侍 seems to be Seven Samurai with no article, though.)
If the quantity is part of the new information to be conveyed, and if you don't need to sound dramatic, then 花を３本見た would be more natural, or neutral.
Choosing the right caption for the linked image is not so straightforward as it seems. ８人の生徒 would seem to give those eight special attention they probably don't deserve, like saying the eight students. On the other hand, 生徒８人 would sound a bit incomplete as a noun phrase, which a caption is expected to be. If I have to include the count, I might settle for the former only because there seems no better option.
８人の wouldn't sound too "specific" if the phrase contained another qualifier as in 笑う８人の生徒. I guess the "specificity" gets diluted compared to when only the number is specifically said.