'Middle Management' is also not a well defined term.
For some, managers like you have described could be thought of as front line managers. Or supervisors.
And middle management are the layers, usually many, that separate the front line managers from the senior executives.
The role of middle management is often to oversee budgets, and be accountable for financial performance in certain geographic regions or business areas.
Middle managers don't employ workers. They employ other managers. Lots of these kinds of jobs will probably be made so simple that entire layers of management will just be removed.
Some examples from chain retail might be:
The franchisee of say, a McDonald's, is in some sense a middle manager. They hire store managers to run the stores in the area that they're responsible for. And they're accountable for the financial performance of all the McDonald's that they own.
The 'Regional Manager' of some large company. They won't be involved in the day to day activity of the business, but they're often an important cog in the wheel because they have local visibility on the financial performance of the business in the region, and are best positioned to allocate the budget in ways that maximize financial performance. Again, they don't hire workers, they hire other managers.