Was it true that Nelson Mandela refused to renounce violence while in Prison?

Indeed, Mandela did not want to renounce violence, during his incarceration, as a means of political change. While he had been offered an end to his imprisonment quite a few times, he categorically refused on the basis that entering an agreement that did not acquiesce to the ANC's demands would be to betray his cause and the South-African people. Take for example, the offer by P. W. Botha on the 31st of January 1985 that offered him conditional release if:

"[Mandela] unconditionally rejected violence as a political weapon." -Pieter Willem Botha, 1985

Nelson Mandela's historic response contained the following:

"What freedom am I being offered while the organisation of the people [ANC] remains banned? Only free men can negotiate. A prisoner cannot enter into contracts." -Nelson Mandela, 1985.

Furthermore, Mandela would not cave in to the Apartheid government in any way. Even on the cusp of victory, he stated in a speech following his release in 1990 from Victor Verster Prison:

"[armed action would continue as] a purely defensive action against the violence of apartheid." -Nelson Mandela, 1990.

Lastly, I would address some misconceptions that are put forward in the cited passage of your question. The assertion begins on a false premise. That Mandela had not suffered under Apartheid and by extension that the whole of non-white South-Africans had not suffered. To say as I did, above, that this is a misconception is charitable at best, to say it is a blatant lie would be truthful. The Apartheid government was responsible for, amongst other things, racist policies that denied the civil rights of South-African citizens, segregating society in South-Africa, criminalising dissenting political parties, censoring the press, deporting non-whites, imprisoning people without trial, aiding and abetting terrorism, torturing prisoners, "disappearing" people, violating human rights, armed intervention in other countries, organising a coup in the Seychelles, massacring its citizens, and assassinating opponents. The fact that the word Apartheid now carries an infamous connotation of segregation and repression underlines the brutality of those years.

Now, turning to Mandela and his treatment by Apartheid. It should be noted that he was imprisoned in three separate establishments with varying conditions: firstly on Robben Island (1964-1982), then Pollsmoor Prison (1982-1988), and lastly in Victor Vester Prison (1988-1990). The first period on Robben Island included solitary confinement, forced labour, a by product of which cost him part of his eyesight, a damp cell with few furnishings, a ban on outside news, abuse by the wardens and very little correspondence with the outside world all contrary to what is said in the article cited in the question. The second period in Pollsmoor Prison did see his general conditions improve with an increase in outside correspondence, the permission to tend a garden, friendlier wardens and better cell conditions. The third period in Victor Vester may be the one that the author of the article you cite is lopsidedly referring to, a period of two of his twenty-seven years spent in prison. This period saw Mandela housed in good conditions, with a personal cook and relative freedom. This last period could be likened to house arrest.

Finally, the absence of any mention in the piece linked in the original of the tuberculosis Mandela contracted due to the conditions of his incarceration highlights the cherry picking and opinion based arguments of that text. Then again a cursory look, let alone an in depth analysis, of the piece reveals just that; it is an opinion piece devoid of historiographic value unless you are studying revisionism.

You may consult the Wikipedia article about Mandela's time in prison, which does mention multiple times his refusal to disavow violent struggle against Apartheid. The article is abundantly sourced from authoritative and balanced references, for your further reading needs.