A recent study by Yamashita and Yasunaga (GameSec 2023) presented a constant-round deterministic broadcast protocol secure against detection-averse adversaries—those who prefer to attack without being detected. In this work, we revisit their protocol and observe that it remains secure even against a broader class of adversaries, not necessarily detection-averse. We formalize its detection mechanism as local detectability and construct broadcast protocols with local detectability that address two weaknesses of the original protocol: (1) it only guarantees weak validity, and (2) it may cause false detections. Our first protocol achieves round complexity four against rational adversaries and \(t+4\) against malicious adversaries, where the adversary corrupts at most t parties. Our second protocol achieves the optimal round complexity of \(t+1\) for malicious adversaries, while the round complexity is four against detection-averse adversaries.