The first commercially successful engine that could transmit continuous power to a machine was the atmospheric engine, invented by Thomas Newcomen around 1712. It improved on Savery's steam pump, using a piston as proposed by Papin. Newcomen's engine was relatively inefficient, and mostly used for pumping water. It worked by creating a partial vacuum by condensing steam under a piston within a cylinder. It was employed for draining mine workings at depths originally impractical using traditional means, and for providing reusable water for driving waterwheels at factories sited away from a suitable "head".
This shows it wasn’t specifically the steam engine that kicked things off, but rather a wide range of factors which eventually caused people to say something different was going on. The stated start of the industrial revolution is really quite arbitrary, but linking it to the seam engine is really a misunderstanding what was going on.
If you want to point to a single invention Benjamin Huntsman developing the crucible technique for steel production in 1740 had a larger impact on the early years of the first industrial revolution. But again it wasn’t any single invention the transition was simply continuous.
England really industrialized via multiple waves transitioning cottage industry and water power driving drop hammers and other equipment to ever more industrial factories. It’s a surprisingly smooth transition where individual inventions like the spinning wheel just keep compounding and driving further increases in efficiency.
Steam power was only used at mines initially because mines where limited by geology. Unlike early factories you didn’t get to decide where the ideal place was you where stuck where the resources where located.