I was taught that I should use " at the start of a quote and if it carries on in subsequent paragraphs, I should put ' at the end of the first paragraph, then start the second paragraph with ' and close again with ' and so on, until the final paragraph which I close with ' ". I cannot see any reference to this style.
I've never seen the style of quoting you mention in use.
Jesse Game-Brown's site states:
The AP Stylebook and The New York Times Manual of Style and Usage suggest beginning every paragraph of a multi-paragraph quote with quotation marks, but saving the concluding quotation marks for the end of the quotation.
It offers this as passage as an example:
“What we, the Tree Warriors, are, despite what we’ve heard, is nothing short of miraculous. Only the most insecure of politicians would dare to attack a group with our integrity and commitment to justice and happiness for all. We are not to be shaken by such words of weakness.
“My assistant, Belinda, tells me that our Tree Warriors association ranks highest among all the associations. Belinda is quite the researcher. Let’s have a round of applause for Belinda.”
This style is what I most commonly see used with multi-paragraph quotes.
Block quotes are usually the clearest way to indicate longer quotations. I've never seen the format you reference.