Communities Secretary Robert Jenrick has insisted now is the right time to update the government's coronavirus message from "stay at home" to "stay alert", amid widespread criticism.
PM Boris Johnson announced the slogan for England, telling people to "stay alert, control the virus, save lives", ahead of his national address.
Scotland's Nicola Sturgeon said: "I don't know what 'stay alert' means."
The first minister added at the daily briefing in Edinburgh: "For Scotland right now, given the fragility of the progress we've made, given the critical point that we are at, then it would be catastrophic for me to drop the 'stay at home' message.
"I am particularly not prepared to do it in favour of a message that is vague and imprecise."
Labour's shadow health secretary Jonathan Ashworth warned people might be "puzzled" by the change.
But Mr Jenrick told the BBC's Andrew Marr: "Stay alert will mean stay alert by staying home as much as possible, but stay alert when you do go out, by maintaining social distancing, washing your hands, respecting others in the workplace and the other settings that you'll go to."
A further 269 people have died in the UK after testing positive for coronavirus, taking the total number of deaths recorded to 31,855.
The number of deaths recorded tends to be lower over the weekend because of reporting delays.
The government has also missed its target of 100,000 coronavirus tests a day for the eighth day in a row, with 92,837 tests on Saturday.
The prime minister shared the new government slogan on Twitter, detailing some of the guidance issued to the public.
Everyone has a role to play in helping to control the virus by staying alert and following the rules.— Boris Johnson (@BorisJohnson) May 10, 2020
This is how we can continue to save lives as we start to recover from coronavirus.#StayAlert pic.twitter.com/2z9yl1Fxs4
A No 10 spokesman added: "Everyone has a role to play in keeping the rate of infection (R) down by staying alert and following the rules."
Mr Ashworth called on the government to clarify what the new slogan meant.
"When you're dealing with a public health crisis of this nature, you need absolute clarity from government about what the advice is. There is no room for nuance," he said.
"The problem with the new message is that many people will be puzzled by it," he added.
Mr Jenrick said the updated message was a "cautious" one, because the rate of infection remained high and the public were "understandably anxious".
He dismissed Mr Ashworth's concerns, saying: "The public are capable of understanding a broader message as we move into the next phase of the virus."
However, the Liberal Democrats' acting leader Sir Ed Davey said changing the slogan "makes the police's job near-impossible and may cause considerable alarm" as he urged the government to publish the evidence that has informed the new strategy.
And on social media, Greater Manchester Mayor Andy Burnham called the updated advice "too ambiguous" and "unenforceable".
The UK government's new slogan is part of moving into the next phase of the response to coronavirus.
Staying at home where possible will remain part of the strategy, but ministers want to "broaden the message".
Some are worried the new campaign is ambiguous and muddies the water.
In Wales and Scotland, the devolved governments who control health have made clear they will keep the original slogan - stay at home, protect the NHS, save lives.
So from tomorrow, messaging will be different in different parts of the UK. And I understand there are real concerns in the Scottish government about how people will react - and fears it will be harder to get them to follow their advice to stay at home unless essential.
First Minister Nicola Sturgeon said she would continue to use the "stay at home" message in Scotland and later said she had asked the UK government "not to deploy" the new slogan there.
STAY HOME. PROTECT THE NHS. SAVE LIVES.— Nicola Sturgeon (@NicolaSturgeon) May 10, 2020
Leaders of the devolved nations - which have the power to set their own lockdown regulations - said they had not been consulted over the "stay alert" message.
Ms Sturgeon said the first she heard of the updated guidance was in newspaper reports.
Giving Scotland's daily coronavirus briefing, she said that, other than allowing people to leave home for exercise more than once a day, the rules there had not changed. "We remain in lockdown for now and my ask of you is to remain at home", she said.
Behavioural expert Professor Susan Michie, who is part of the government's Scientific Advisory Group for Emergencies (Sage), warned some people might take the more generalised "alert" slogan as a "green light" to socialise.
"I do not think this is a helpful message in terms of guiding behaviour. It does not give advice as to what people should do," she told the PA news agency.
Meanwhile, the government's pandemic response has been called "wishy-washy" by a body representing police officers in London.
It comes after a police force in east London shared an image of a crowded park in Hackney on Saturday, where hundreds of people, they said, were eating and drinking alcohol.
Ken Marsh, from the Metropolitan Police Federation, told BBC Radio 4's Today programme authorities "needed to be firmer right from the beginning" and that if authorities had been more stringent from the outset "we would have a better result now".
But another adviser to Sage, Prof Mark Woolhouse of Edinburgh University, told BBC Radio 4's The World This Weekend outdoor activities posed a "relatively low risk" so long as people with symptoms were not going out and that those who needed to quarantine themselves did so.
In his address, the prime minister announced the launch of an alert system for tracking coronavirus in England and unveiled a "conditional plan" to reopen society, allowing people in England to spend more time outdoors from Wednesday.
Mr Johnson also said people who could not work from home should return to the workplace - but avoid public transport.
The lockdown has already been extended for another three weeks in Scotland, Wales and Northern Ireland to 28 May.
In other developments:
- The Department of Health has confirmed 50,000 coronavirus test samples were sent to the US earlier this week after problems in UK laboratories. It comes as the government failed to hit the 100,000 daily testing target for the seventh day running
- More than 70 public figures - including Baroness Doreen Lawrence and author Malorie Blackman - are calling for a full independent public inquiry into deaths from Covid-19 among people from ethnic minority backgrounds