Just to follow up on Chris-san's explanation... I'm not exactly sure what's confusing you here.
Both of the sentence pairs are identical, in that sentence #1 of each describes performing the second action having done (after doing) the first, and sentence #2 describes performing the second action without having done the first.
Your confusion seems to arise from the English translations you're assigning to both of the sentence #1's. Translation can be helpful to check as to whether you fully understand the meaning of a Japanese phrase or sentence, but you shouldn't rely on translation to interpret/understand Japanese grammar, because many times the most natural English translation of a given sentence will not correspond to the Japanese sentence structure (and there are times where certain nuances are not effectively translatable).
One thing that perhaps might be confusing you is that the Japanese verb 持つ by itself doesn't mean "take" as in "bring something from one place to another" but literally "take into hand". If I see an umbrella in an umbrella stand, reach out, grab it, and lift it up, I could describe what I just did as 傘を持った (and, of course, my current state as 傘を持っている). 食べる, on the other hand, maps quite closely to the English "eat".
Maybe that can help you see how both of the sentence #1's are describing the same exact sequence of events: A person taking an umbrella into hand and (then) going out (with the umbrella in hand), and a person eating breakfast and (then) going out (with the food in his/her belly).