So a Beech 1900 with 19 seats (duh) operating as a scheduled airliner would be Part 121 or 125? Talk about headspin.
There is a concept of "holding out" in the FAA guidance which you can think of as advertising. If a company says, "We have a flight at noon from ABC to XYZ and tickets are $100" they are holding out to the public to provide common carriage between ABC and XYZ. That would fall under 121. A company may have facilities at many locations and have their own air operations to shuttle employees between those locations. General Motors used to do this out of Detroit with CV-580s. But these flights were not open to the general public. They fell under 125.
So in your example "operating as a scheduled airline" means it would fall under 121. You won't hear about much 125 stuff unless you are part of that operation since by definition they can't advertise.