Barnaby Raine, a Jewish Labour party member and supporter of Mr Corbyn, said it marked a significant step forward for the Labour leader, adding he had not only condemned people who hold anti-Semitic views but also those who "dismiss this whole thing as just a smear on the Labour Party".
A spokesman for the Labour Friends of Israel group said Mr Corbyn's words lacked credibility so long as he refused to apologise "for the many occasions on which he himself has offended the Jewish community".
Speaking on the BBC Radio 4's Today Programme, Ms Cooper said: "It is terrible for the Labour party, that's got such a history of fighting racism, to be stuck in this mess".
The party was reported to be ready to adopt three of the internationally accepted definitions of anti-Semitism which it had previously refused to accept.
Supporters of Corbyn alleged Watson's comments, which he made in an interview with the Observer, came after a dispute with the leader's office over a speaking slot at the Labour Party conference.
The Labour MPs, who both lost family members in the Holocaust, are facing disciplinary proceedings for confrontations over the party's response to anti-Semitism claims.
"Our party must never be a home for such people and never will be".
Walthamstow's Stella Creasy: "If you are calling for Tom Watson to resign because he's upset you wanting to tackle racism within the Labour Party so that it doesn't stop us winning an election or living our values as Labour, you've spent too long in the midday sunshine".
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Former foreign secretary David Miliband offered Mr Watson his support, saying: "Thank you Tom Watson for recognising the gravity of Labour's situation".
'And anyone who denies this has surfaced within our party is clearly actually wrong and contributing to the problem'.
Corbyn has been battered by criticism in recent years over what critics have said is the opposition party's failure to address anti-Semitic expression in its ranks. But he said there was an issue about "half of one example" because it could restrict criticism of Israel.
Mr Corbyn issued a video message apologising for the "hurt" caused. "We're acting to speed this process up".
The dispute centres on the decision by Labour's ruling National Executive Committee not to adopt into its rulebook the full International Holocaust Remembrance Alliance (IHRA) definition of anti-semitism, including its 11 examples.
Gideon Falter, chairman of the Campaign Against Antisemitism NGO, responded to Corbyn's video by saying it was "devoid of any apology for his own antisemitism or promises of specific actions", and added adamantly that "Jeremy Corbyn is an antisemite and under his leadership the Labour Party has become institutionally antisemitic and an existential threat to British Jewry".
Sunday's video comes after Mr Corbyn wrote a piece in the Guardian on Friday, in which he resisted the same demand over the IHRA definition from others.