Увидеть is a perfective verb.
You can use увидеть to mean the act of registering something visually. You might have heard that perfective verbs are essentially about the localisation of an action in time, associated with a transition point between two states (e.g. «заснуть» to fall asleep: not asleep → asleep).
With увидеть, you go from not seeing something to seeing it. To arrange actions into a sequence, you need a string of perfective verbs.
- you can, actually, use "видеть" in this sentence. It would mean that видеть is a framing action (you were in a state of seeing everything, and made notes of it when you had time)
- you can also use the imperfective записывал—then the actions are not localised at all. Such a sentence would mean that you habitually wrote down everything that caught you attention (during the time period you are talking about).
Написать is the more-or-less neutral perfective of "write". It means scribbling words on paper or typing them (or creating music; in Russian писать also extends to painting with oils).
Записать is "to record", "to write(note) down". That's quite a good correspondence. When you record something, you do it to store the information (and then you can access that indormation later). It is also applicable even if you are not the source, the author of said information; you can "make notes" of someting another person told you, but you can hardly say that you "wrote" it.
In English, "to write down" carries with the idea that you are taking careful note of the source of what you're writing, and you are making a strong effort to record it accurately - even if the source is the thoughts in your own mind. "Mozart wrote down the music that he had composed in his head" "As Mozart lay dying, Salieri wrote down the music that Mozart was dictating to him."