The UK Brexit minister David Davis tried to play down the row on Wednesday, telling a parliamentary committee in London that despite some "sharp" remarks, the European Union president had emphasised that Britain was a friend.
The votes came after Brexit minister Lord Bridges of Headley warned now was not the time to "return to the fray" by inserting "terms and conditions" in the legislation.
"We have been clear that the prime minister will trigger Article 50 by the end of March", her spokesman said, heavily emphasising the word "end".
But May succeeded in warding off a potential rebellion from a handful of pro-EU Conservatives in the lower chamber, the House of Commons, where May only has a slim majority.
Scotland voted overwhelmingly (62 per cent to 38 per cent) to remain in the Brexit referendum - as did Northern Ireland, while England and Wales mostly voted to leave.
Once the council at its meeting in May has provided a broad political response to the United Kingdom, along with a framework for the talks, known as the guidelines, the European commission will then recommend to the council that the negotiations begin. The government argued against the changes, saying it wanted to preserve May's flexibility in the talks.
Up to 10 Tory MPs are reportedly considering opposing the government or abstaining in the vote but a rebellion would need to reach higher numbers to derail the process.
"We will not be intimidated by threats and I can assure you they simply will not work", Tusk, the chair of European Union leaders' meetings who will play a key role in Brexit talks, said.
She will ask the Scottish Parliament next week to request a Section 30 order from Westminster.
Announcing Britain's formal intention to leave poses a risk for May as it cedes power to the EU.
"We are determined parliament will be engaged all the way through the process and afterwards", he told reporters.
Just as May embarks on Brexit talks, she faces another crisis at home.
First Minister Nicola Sturgeon, the leader of the Scottish National Party (SNP), has said Brexit makes a second independence vote "highly likely".
Tusk also reiterated his concerns over plans to allow some of the remaining 27 European Union states to foster closer ties after Brexit, leaving those more reluctant member of the bloc behind.
Independent crossbencher Lord Pannick, who represented lead claimant Gina Miller in the successful Article 50 legal challenge and moved the original amendment on holding a "meaningful" final vote, said he regretted the Commons had not taken the advice of the Lords.
Republican party Sinn Fein has been urging a referendum on Northern Ireland leaving the United Kingdom since the Brexit vote a year ago.