Family members who have set up camp outside the mine shaft watched the video on cell phones and a screen erected by the government.
The video showed around a dozen miners at times waving and laughing, and at one point arm-in-arm, singing Chile's national anthem and chanting "Long live Chile, and long live the miners".
One miner sent greetings to his relatives.
"I'd like to say hello to my grandchildren and all my family. Stay together," he said.
First lawsuit filed
Also on Thursday, the family of one of the trapped men announced it would sue the San Esteban Mining company, which owns the mine.
The family's suit alleges that San Esteban negligently allowed the mine to reopen in 2008 after being closed the previous year due to an accident, the family's lawyer, Remberto Valdes, told reporters.
The suit names the owners of San Esteban, Alejandro Bohn and Marcelo Kemeny, as well as the government's National Mining and Mines Service, which approved the reopening.
A judge ordered San Esteban to freeze $1.8 million in anticipation of more suits, but the company has said it may go bankrupt and cannot pay its workers.
Chile's state-owned mining company is working on the rescue at an estimated cost of around $1.7 million.
Months before rescue
The miners have been told they may be stuck underground for months before rescuers can free them.
|The government has warned the miners may begin to suffer from depression [AFP]
Officials on Wednesday said it could take up to four months before the men can be freed and that, until then, they will get oxygen, food, water and medical supplies through three thin, newly drilled tunnels.
The news was delivered as the government, consulting with submarine commanders and the US National Aeronautics and Space Administration, prepares a special programme to help the miners cope mentally and physically with their prolonged captivity.
Chilean engineers have said they need at least 120 days using a hydraulic bore to dig a narrow escape shaft measuring just 66 centimetres in diameter, or roughly the size of a bicycle wheel, to get the men out.
The mine runs like a corkscrew for more than seven kilometres under a barren mountain in northern Chile's Atacama Desert.