The messenger (an angel, which is clarified in verse 17) declares his authority to Alma. In addition to declaring his authority (having been sent from God) he also reminds Alma of the proof of his power. This proof is to remind Alma that he is first of all being seen at all, and secondly that his voice shook the earth. Note the discussion above concerning the possible Mesoamerican significance of both the voice and the shaking.
The angel explicitly uses those two markers as the proof of his divinity. While the presence of the angel does not mention the cloud, the original description of the vision explicitly places his person in the cloud. Thus the angel very specifically repeats to Alma the three markers of his divinity that were first noted; the cloud, the voice as thunder, and the shaking of the earth. The repetition and highlighting by the angel should confirm to us that the specific markers are intended to be significant, and not simply a description of the appearance.