There is a lot to unpack here, and I may not be the best person to unpack it, but I believe I can help put you on a better path.

So in general, marxist history (or historiography) means viewing history primarily in economic terms as a succession or evolution of economic systems or stages. It is not at all the same as Marxist political views, although sometimes they overlap. Even a rather right wing author like Niall Ferguson who emphasizes economics in his works might be said to be borrowing ideas from marxist history. (On the other hand, some of the more commonly read marxist historians also hold left of center political views) While marxist historians do emphasize "history from below" and the experience of the "common man", this alone wouldn't necessarily be marxist history without the emphasis on economic systems and relations.

Marxist historical theory is generally regarded by historians as a useful perspective, especially in economic issues, and is not nearly as much of an outlier or outcast as Marxist political views are in Britan and the United States. Marxist history is also criticized for being somewhat deterministic (i.e. History is expected to have an end goal) and neglectful of non-economic factors, but it's very much part of mainstream academic discourse.

As for Howard Zinn, while his politics may lean left, he is not a marxist historian in the sense described above, because he doesn't really focus on economic relations. As a political science professor by training, Zinn isn't really doing history the way any kind of historians do it, but crafting a political polemic. A better example of a left leaning person doing actual marxist history might be someone like Perry Anderson; his Passages from Antiquity to Feudalism can show up in an undergraduate history course without much controversy.

To give a concrete example, no one would dismiss Eric Hobsbawm but he was a Marxist not just in personal politics but also in his historical analysis. His work is carefully researched and often brilliant. He happened to believe that the best explanation for the evidence he presented came within a Marxist framework. Many people who love his work disagree with that analysis but it is a serious discussion. Theory can act like framing a picture, it allows the historian to ask a set of commonly understood questions and interpret the answers. The problem is when the answers don't fit in your frame it is tempting to squash them into shape.

Almost all current theoretical approaches that historians use are more or less descended from some of Marx's ideas. This is sometimes straightforward economic history (where social change is driven by infrastructure), or a more radically political methodology like Zinn. But ideas that humans shape and are shaped by (at the same time) their environment, that material objects can communicate cultural values, that people's social groups influence the things they like and do - these are all indebted to Marx's ideas. In some cases, this influence is buried under so many layers that we don't talk or think about it very often. But it's foundational.

Many people will use 'Marxist' as an insult, especially to dismiss overly simplistic analyses that assume everything can be explained by the price of bread, or that ideas don't really matter so much as following the money. That's only a sliver of Marx's influence, however, and I'm personally very happy to acknowledge my intellectual debt his ideas.

And, I should add, the problem with categorizing one specific Marxist approach is that Marx didn't restrict himself to a single historical approach in all of his writings. Some of his work emphasizes the important role of ideas in shaping and perpetuating class structures; other times he focuses more on economic infrastructure. This isn't a contradiction - it just means there are many ways to engage history, and Marx played around with enough of them that it would be very difficult to escape lines of inquiry that he helped shape. And Marx's best work looked at the relationships and tensions between these different kinds of social structures, which is a method historians today almost universally respect and use.