document.body.className = 'class1 class2'
[0] https://dom.spec.whatwg.org/#ref-for-dom-element-classname%E...
And the DOM spec changed these names because "class" and "for" are reserved words in Java (and if you look closely, you'll notice that the main target language for the DOM spec was Java). If you're writing a template language, you don't need to be bound by reserved words in another language, and matching the original attribute name is more intuitive than matching the underlying DOM implementation.
(But I can understand why JSX did that: since it maps directly to DOM calls, using the same name as DOM avoids the need for a table mapping from the attribute name to the DOM name. That is, implementation convenience was more important than usability.)