Your understanding is in the ballpark. APIs were one tool in the SOA toolbox. Except you are seeing APIs as just technology.
APIs have jumped out of the enterprise toolbox, and found success in the richer oxygen environment of the Internet, escaping from the claustrophobic environment of the restrictive enterprise network in which strict governance was imposed, and technologists decided everything.
After going outside the firewall, APIs became about the simplicity of REST + JSON over HTTP, putting these resources closer to actual problem owners--flipping the governance of SOA on its head, making APIs more about partnering, collaboration, transparency and openness, and not just about control. After escaping the governance and the bottleneck of traditional IT, this new breed of APIs allowed for new types of innovation, business models and opportunities amongst not just open developers, but actual business owners. A new world emerged that the governance overlords of SOA could never achieve or even see.