From my perspective a programmer is just able to write code, i.e. turn the business requirements into instructions.
A software engineer is rather able to design/architect a system, let it be a single web page or a container cluster.
Then the question is how people learn "how to architect a complex application"?
Once you can write code you can architect a system. There’s nothing stopping you. The information is out there.
Perhaps the real question is whether your company trusts you enough to make something from scratch. Or whether your solution is good enough to handle large amounts of customers, traffic, or usage.
People learn to do so with a combination of experience and just general learning (documentation, tutorials, references, talks, conferences, etc) over time.
I think a modern company would simply call both roles “software developer” and apply different levels of seniority (e.g. junior, senior, staff, or they could use numerical levels).