偏見: appearance-based prejudice and awareness of bias
Distinguish wanting to remove prejudice from recognizing one's own bias through travel, without treating either as completed removal.
The basis for judging someone
偏見 (へんけん) means prejudice or bias. “外見だけで人を判断する偏見をなくしたい。” expresses a wish to overcome prejudice that judges people by appearance alone.
外見だけで limits the basis of judgment to appearance. The example supplies neither a prejudice about a particular group nor proof that the appearance-based judgment was accurate.
Becoming aware of one's own bias
“彼女は旅を通じて自分の偏見に気づいた。” says that travel made her aware of her own biases. 自分の identifies the bias she recognized as her own.
気づいた states that awareness occurred. It does not say how much her views changed afterward or that all of her bias disappeared.
A wish and an accomplished act of noticing
なくしたい expresses wanting to remove something, while 気づいた reports becoming aware of it. The two sentences show different stages in relation to prejudice.
A wish to remove bias is not completed removal, and awareness is not proof of having no bias left. Keep the extent of change actually stated by each sentence.
Does becoming aware of her biases through travel mean that she removed them all?
Show the answer and why
No. The sentence reports awareness of her own biases, not their complete removal. The other example's なくしたい also expresses a wish rather than an accomplished result.