冷静 and 冷淡: composed judgment and a cold response
Compare answering an unexpected question calmly with responding coldly to a request for help. Distinguish a proposal to wait for composed judgment from a response that actually occurred.
Composure in answering and judging with 冷静
冷静 (れいせい) describes being calm or composed. 「予想外の質問にも冷静に答えた。」 describes a composed answer even to an unexpected question, without establishing whether the answer was correct or warm.
「まず冷静な判断ができるまで少し待とう。」 proposes waiting a little until a levelheaded judgment can be made. Read 「できるまで」 together with 「待とう」: this is a proposal to wait for that ability, not a report that a judgment has already been completed.
The attitude toward another person with 冷淡
冷淡 (れいたん) describes a cold or indifferent attitude toward someone. 「助けを求める声に冷淡な態度を取った。」 reports such an attitude in response to a voice asking for help.
「彼の返事は驚くほど冷淡だった。」 describes his reply as surprisingly indifferent. It does not explain his reason for replying that way or specify what help was provided or refused.
Keep composure distinct from indifference
The question example concerns the composure of an answer, whereas the request-for-help example concerns the attitude shown toward another person. Translating both as a cold manner would blur the distinction between calm judgment and an indifferent attitude.
A composed judgment need not be indifferent, and a cold reply may also be delivered calmly. Allow these features to coexist without using an unstated motive or an invented outcome of the request to distinguish the words.
Does showing 冷淡な態度 toward a request for help necessarily mean that help was actually refused?
Show the answer and why
No. 「冷淡な態度を取った」 describes the attitude shown in response to the request. It does not specify the concrete outcome of providing or refusing help, or the reason for that outcome.