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The condition of quasi-compactness in the Zariski topology bears little resemblance to the condition of compactness in the classical analytic topology: e.g. any variety over a field is quasi-compact in the Zariski topology, but a complex variety is compact in the analytic topology iff it is complete, or better, proper over $\operatorname{Spec} \mathbb{C}$.

I think many algebraic geometers think to themselves that a variety is "compact" if it is proper over the spectrum of a field. I have heard this terminology used and occasionally it shows up in (somewhat informal) writing.

So a perhaps more accurate brief answer is that in algebraic geometry the distinction between quasi-compact and quasi-compact Hausdorff is very important, whereas in other branches of geometry non-Hausdorff spaces turn up more rarely.

Anyway, many mathematicians have been happy with the quasi-compact / compact distinction for about 50 years, so I don't think this usage is going away anytime soon.

To address the last question: when writing for a general mathematical audience, it is a good idea to give an unobtrusive heads up as to your stance on the quasi-compact / compact convention. (The same probably goes for other non-universal conventions in mathematics.) If I were speaking about profinite groups, I would say something like:

"A profinite group is a topological group which can be expressed as an inverse limit of finite discrete groups. Equivalently, a topological group is profinite if it is compact (Hausdorff!) and totally disconnected."

This should let people know what side I'm on, and thus be able to understand me. When writing for students, I might take pains to be more explicit, using a "By compact I mean..." construction as you have indicated above.

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The condition of quasi-compactness in the Zariski topology bears little resemblance to the condition of compactness in the classical analytic topology: e.g. any variety over a field is quasi-compact in the Zariski topology, but a complex variety is compact in the analytic topology iff it is complete, or better, proper over $\operatorname{Spec} \mathbb{C}$.

I think many algebraic geometers think to themselves that a variety is "compact" if it is proper over the spectrum of a field. I have heard this terminology used and occasionally it shows up in (somewhat informal) writing.

So a perhaps more accurate brief answer is that in algebraic geometry the distinction between quasi-compact and quasi-compact Hausdorff is very important, whereas in other branches of geometry non-Hausdorff spaces turn up more rarely.

Anyway, many mathematicians have been happy with the quasi-compact / compact distinction for about 50 years, so I don't think this usage is going away anytime soon.