Could you go a bit more in details of what you mean here? What strange things have you seen?
"Small business" is weird term.
The Small Business Administration defines a "small business" as one with revenue between $1 million and $40 million, and between 100 and 500 employees. To my mind, no business on that scale is a "small business".
As a result, for decades, I've considered "small business" to be essentially a meaningless term.
A startup is a specific sort of thing, and may or may not be a small business (even by the SBA's definition). I've long avoided labelling my startups as a "startup" because that term comes loaded with all sorts of implications I'd rather avoid.
I do what you mentioned: my business is a "business". No adjectives are necessary.
People who are starting a business will just start a business and call it a business. People who are marketing themselves as a small business right now, and comparing it to a startup, seem to be a weird third group.