侮辱 and 軽蔑: a perceived insult and an attitude of contempt
Compare who perceives a remark as an insult with what prompts or expresses contempt. Separate a character's interpretation from proven intent, and allow contempt to appear in words.
Who considers what an insult
侮辱（ぶじょく） means insult. 彼女はその発言を仲間への侮辱だと感じた。 says that she felt the remark was an insult to her colleague. She is the person interpreting it; the colleague is its perceived target.
相手を侮辱する言葉を使わずに意見を述べた。 describes expressing an opinion without using words that insulted the other person. Read the stated action together with the language that was not used.
Feeling and expressing contempt
軽蔑（けいべつ） means contempt or scorn. 弱い者を笑う態度に強い軽蔑を感じた。 describes strong contempt toward the attitude of laughing at the weak.
相手を軽蔑するような言葉は議論を壊してしまう。 describes words suggesting contempt for the other person as damaging a discussion. Contempt can appear in language, rather than remaining only an inner feeling.
An interpretation is not confirmed intent
侮辱だと感じた reports how a character understood the remark. It does not independently confirm the speaker's intention, so keep the person interpreting the remark separate from its speaker.
Do not assign 侮辱 only to spoken words and 軽蔑 only to private feelings. Identify the remark or attitude, who reacts to it, and how that reaction is expressed.
Does 彼女はその発言を仲間への侮辱だと感じた。 confirm that the speaker intended to insult her colleague?
Show the answer and why
No. It reports her interpretation of the remark as an insult to her colleague. Confirmation of the speaker's intention is a separate matter.