諦める and 失望: what is abandoned and what disappoints
Compare abandoning today's outing because the rain continues with disappointment at a canceled exhibition. Preserve the limits of the decision and the statements that one failure need not lead to giving up or self-disappointment.
What is given up with 諦める?
諦める (あきらめる) means to give up or abandon an action or goal. In 「雨がやまないので、今日は外出を諦めた。」, the speaker gave up on going out today because the rain did not stop; 「今日は」 limits the scope of that decision.
「一度の失敗で夢を諦める必要はない。」 says that there is no need to abandon a dream after one failure. Preserve 「必要はない」 rather than changing the sentence into a report that someone failed once and already gave up the dream.
The result and target of disappointment with 失望
失望 (しつぼう) describes disappointment when expectations are not met. In 「期待していた展示が中止になり、皆が失望しました。」, the anticipated exhibition was canceled and everyone was disappointed.
「一度の失敗で自分に失望する必要はありません。」 says there is no need to be disappointed in oneself after one failure. The target is oneself, and 「必要はありません」 does not report that this disappointment actually occurred.
Separate a decision to stop from a feeling of disappointment
For the outing, identify the abandoned action and its limit to today; for the exhibition, identify the cancellation and the people's response. Giving up today's outing does not establish permanent abandonment of all outings or of a dream.
Someone may give up an activity while feeling disappointed, but disappointment alone does not establish a decision to stop. The two failure examples say that abandoning a dream or feeling disappointed in oneself is unnecessary, so neither outcome is an inevitable consequence of one failure.
Does 諦めた in the rainy-day example mean that the speaker decided never to go out again?
Show the answer and why
No. The decision concerns going out today because the rain continued. Later outings and other goals are not stated to have been abandoned; keep the scope marked by 「今日は」.