Não, aquele site não lida com todos os pormenores. Sim, nesse caso que você menciona, o padrão é recomeçar com aspas, e depois uma maiúscula.
As aspas são a parte mais importante, pois definem aquilo que deve ser atribuido a cada personagem do diálogo.
Muitos sites lidam mais com as situações em que se usa "he said", "she said", explicitamente. Mas muitos diálogos -- pra não ficar chato -- utilizam outros artifícios, e contam mais da história ao mesmo tempo. Um exemplo:
Richard walked in and surveyed the scene. "Hi, Linda. How nice! We have guests!"
His wife looked up and smiled. "Oh, sorry, honey, I had no way to tell you Mary was coming over." She motioned toward the guests. "You remember Mary from the other night? This is her brother John."
"How could I forget. She saved our lives. Hi, Mary! Nice to see you again." He stepped into the room toward the visitors. "Don't get up John," as they shook hands.
A momentary pause descended on the group, as if some new uncertain topic waited to be launched.
"I asked to come by," said Mary hesitantly, "because I have some strange news about the other night. I'll let John explain."
The brother shifted uneasily in his chair, looking from Linda to Richard. "You don't know this, but I am a private investigator." He waved toward Mary. "She helped you out more than any of you can imagine. What was happening the other night was not an accident." He looked at Richard for a full moment. "You were being targeted, Mr. Smith."
The point is, you can tell who's talking with the help of other descriptive details, etc., and with the right punctuation. And keep the story moving along without a lot of "he said/she said". Paragraphing and punctuation can help a lot.
(At least I hope you were able to tell, because writing good dialogue is really hard!)