I think it predated Smith. Someone (Lights I think) said something to the effect of expecting him to have a voice like a foghorn because of that stereotype.
Very true. Here is the extract from Lightoller's book.
"Captain Smith, or 'E.J.' as he was familiarly and affectionately known, was quite a character in the shipping world. Tall, full whiskered and broad. At first sight you would think to yourself, "Here's a typical Western Ocean Captain. Bluff, hearty, and I'll bet he's got a voice like a foghorn." As a matter of fact, he had a pleasant quiet voice and invariable smile. A voice he rarely raised above a conversational tone. Not to say he couldn't, in fact, I have often heard him bark an order that made a man come to himself with a bump. He was a great favourite, and a man any officer would give his ears to sail under. I had been with him many years, off and on, in the mail boats."
There were quite a number of references to Captain Smith speaking quietly and calmly, but also moments when he barked out orders during the evacuation. e.g.
Mr. Jones - "The Captain shouted again, in fact, twice again, "Any more ladies?""
Boxhall - "The Captain shouted through a megaphone and ordered our boat to come back."
Lightoller - "I later saw the Captain using a megaphone. He was ordering the lifeboats to come back. I heard him two or three times hail the lifeboats through the megaphone to return back to the ship."
Mrs. Douglas - "Just before we got into the lifeboat the Captain called out, "How many of the crew are in that boat? Get out of there, every man of you.""
Samuel Hemming - "The Captain yelled out, "Everyone over to the starboard side, to keep the ship up as long as possible.""