Is there a way to tell what has caused a 'Check Engine' light to come on without a diagnostic tester?

No way that I'm aware of. A "diagnostic tester" will not give you the magic answer. It will give you a code that narrows down the problem area. It does not tell you what the exact cause is. For example, You get a code that says O2 Sensor 2 bank 2 lean condition. That shouldn't be interpreted as "replace oxygen sensor" Autozone will generally tell you that though. An exhaust leak will provide more oxygen and show a lean condition among others.