Branding is important.
The Liberal brand is popular, even when the Party, the Leader, and the Policy is not. People identify themselves as "Liberal", even if they are not happy with the Liberal Party.
The Conservative brand is similar, in that people can say they are "conservative", despite what the Conservative Party may be doing.
To say you are a "New Democrat" ties you to the NDP. You are not going to call yourself a "New Democrat" if you don't like the Party, Leader, or Policy.
I think, however, the term "Democrat" does speak louder about a self-identification than any particular Party. I could say I was a "Democrat" and support any of the three parties above. Indeed, I have, and I do.
In terms of the US politics, most Canadians would vote Democrat over Republican. For one thing, Canada is not a Republic, so it would be hard to identify with "Republican" without being disloyal to the concept of the Canadian State.
So I think "Democrat" would be the best brand for a new party uniting the anti-Conservative. For one thing, any party opposing the Canadian Democrats would have to explain how they could be democrats without being Democrats.
To win, any party must capture the Canadian centre, which the Conservatives seem to be doing the best at currently.
Until then, I think Jack should stress the Democrat over the New. I also think the NDP should be the lead party in the new coalition, just as the Alliance was the lead party in the coalition on the right.
If they were to call themselves just the Democrats, I would not be surprised if I was joined by Conservatives and Liberals in wishing this new party well.