New Left, because I'm not a racist class reductionist.
Didn't the Old Left support the Civil Rights movement?
Yes and the descriptions of the OP are oversimplified and not accurate whatsoever.
All he said was that the Old Left was "more class reductionist," vis-a-vis the New Left which is presumably "more intersectional." Ignoring the fact that "class reductionist" has become something of a slur, one can still acknowledge the class-based reality of the Civil Rights Movements (as many of the Civil Rights leaders themselves did) and support it on that basis, as all credible leftists of the time did.
Not really. Let’s break this down further.
What are the differences between the two? Some of these are subjective, but some high-level observations…
Industries: Old Left more trades, New Left more services
Immigration: Old Left more skeptical of immigration
Environment: New Left more hawkish on climate
Internationalism: Old Left was more isolationist
Identity: Old Left was more class reductionist
In general it seems the power base has shifted from Northeast and Midwest to Acela and I5.
The Old Left also viewed employers as adversaries. The New Left is a lot more corporate-friendly.
Immigration was a less mainstream issue when the “Old Left” was in power to a large extent compared to today but given “they” inacted the 1965 immigration act and the seeds to the Old Left voted against the 1923 immigration act. Industries really depended on the time and region but Harry Truman in his 1948 DNC speech said Democrats would tear down “trade barriers”. Climate change was a niche issue until the 80s and 90s but in the northeast at least pro-environment has been a constant among Democrats for a century. Considering the New Left emerged in part due to opposition to the Vietnam War, the foreign policy point is very difficult to reason with. The rest of the post I agree with decently but is still probably an oversimplification.
But what part of that has to deal with the Civil Rights movement? Even if OP was accurate in his definition of Old Left, the VRA has little to do with environmentalism, feminism, immigration, foreign policy, trade policy, or whatever other pet issue there is. So there's nothing about OP's definition, however faulty it may be, that would indicate the Old Left would not support Civil Rights.
As you mention there is no perfect definition of the "Old Left" or "New Left" position, because like anything else, it varies from time to place. However, the Old and New Left are just that:
Left. So any iteration of the Democratic Party would not count because the Democrats were at no point ever trying to overcome capitalism and replace it with socialism. The most left-wing Democratic president, Franklin Roosevelt, went no further than Keynesianism. You could call Roosevelt, Truman, and Johnson representatives of "old guard liberalism," but they were not leftists of any kind.