The first problem is that you're reading a textbook. Textbooks are not ways in which historical research is reported; they're primarily teaching tools and are highly criticised and considered bad for teaching in some systems.
Your textbook gives us some clues about how the authors are using "class," a complex theoretical tool.
as Marx stressed…middle-class and peasant property owners feared the socialist challenge of urban workers
We can unpack this.
There's an offhand reference to Marx. Marx views social class as determined by a person's relationship to the means and tools of production in capitalism. For Marx:
- The middle-class owns capital, they are "the bourgeoisie," and their nearest analogue in today's society could be familiar to you as "the 1%": those people who own or control the productive capital of society. In Marx's day this class was larger, and owned smaller companies or factories.
- The peasantry who "own property," the text is referring to occupying owners of agricultural land who still predominantly work their own land despite using levels of hired help.
- Urban workers. This is problematic. Marx and Engels certainly concentrated on the industrial and productive portion of the proletariat, or "working class," but their category wasn't "urban workers." So we have a clue that your text book isn't using Marx's categories but a different set.
The other clue is in "French politicians represented special-interest groups, particularly middle-class ones." where classes are depicted as (primarily) segmented interest groups. Not as relations to production, but as ideas groups.
We can assume that the language being used is untheorised identification of past self-representation by status group, "the middling classes" etc. Here "middle class" would mean, rather than those people who owned capital and subsisted off extracting surplus value, it would mean people displaying the culture of the urban burgher and the rural farmer: people who incidentally owned capital, but who could be identified by their tight closed sitting rooms, control over parliamentary politics, inheritance strategies.
More Bourdieu or Weber than Marx.
I thought that the middle class was the urban working class.
This is a conceit almost entirely confined to US internal propaganda, and a particular kind of instrumentalist sociology that likes to divide populations based on decile groups of income received. There will always be that 40-60% of income receivers who earn greater than the bottom 40% and less than the top 40%. This is not a coherent theoretical tool.
What were some of the occupations of the middle class?
Using Zola for second empire France: financial speculator, manufacturer, wholesaler or mass retailer, rentier, government minister, politician, medium and small retailer, farmer (ie: production using labour for profit without working the land oneself), certain kinds of priest, banker, officer (above certain ranks by mid career), food market wholesaler, wife, scientist, unmarried daughter, expert whore, coal mine owner.
Would business owners be considered part of the middle class?
In the textbooks' understanding of class: yes, all except owner-operator peasants and the most destitute small shop owners or stall holders.
I do not believe the textbook to be correct, useful, or representative of the state of research.