I'd say thinking like a programmer would mean something along the lines of "modeling out the problem".
It also might mean thinking about the current problem, with a bit of an eye towards the future problem (but not too much!)
Otherwise I don't know, I'd hate to think this question could lead to telling people what "types" of thinkers make for good computer programmers. Thinking like a programmer can mean highly logical or highly creative and abstract.
I'd perhaps say that systems thinking (whatever exactly that means) is a way of thinking like a programmer. An operating system, an API infrastructure, a program, it's all a "system" I think.
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