[robotresearcher]

The OED keeps track of original usage. Here's the entry:

orig. U.S. A business or enterprise that is in the process of starting up, or that has just been established; a startup company, venture, etc.

They cite these early usages in print:

1975 Chicago Defender 20 Sept. 23/1 The many problems faced by any new start up, but most assuredly by a black financial institution.

1986 Your Business Mar. 11/3 There are the small businesses; the heroic little start-ups in their garages and garden sheds.

[Nokinside]

There is difference of start-up and start-up company.

Start-up company in the modern sense was first used in 70's