Lord Hall said: "He was an engaging, authoritative and sometimes mischievous broadcaster, who commanded the respect and love of his colleagues and the public alike. He will be hugely missed."
Ms Williams added: "When you heard those deep tones on air you could expect the best - you believed what he said - and you felt, somehow, safe. He was enormously kind to me when I was a young producer and we would often share a chat over the headlines late at night in Broadcasting House. We will miss him."
Eric Coates, 69, who attended Woolverstone Hall with Donaldson, said he had been "quiet but very confident" at the school.
Jeremy Vine, the Radio 2 presenter, tweeted: "Peter Donaldson, who has died, was a brilliant BBC newsreader."
Jamie Angus, the editor of Today, tweeted: "He was quite simply the voice of Radio 4 for a generation."
Martha Kearney, who presents the BBC's World at One programme on Radio 4, tweeted: "So sad to hear about the death of Peter Donaldson. He was the newsreader when I started on WATO and was so kind."