Necessity of being Hausdorff in the definition of compactness?

I would argue that, overall in modern mathematics, the accepted meaning of "compact" is "every open cover has a finite subcover", i.e., it does not include the requirement of being Hausdorff.

The approach of including Hausdorff-ness in the definition of compact, and instead using the word "quasi-compact" for the less restrictive condition, is traditionally associated with French mathematicians and algebraic geometry. See for example Compact and quasi-compact, and quasi-compact and compact in algebraic geometry, and French notational differences.

There are many examples. The simplest is any space $X$ with more than one point that has the indiscrete topology, $\{\varnothing,X\}$. The simplest $T_1$ examples are any infinite set with the cofinite topology. The line with two origins is another $T_1$ example.

Many of us topologists feel that there is no good reason to include Hausdorffness in the definition of compactness and prefer simply to add the requirement and then talk about compact Hausdorff spaces when Hausdorffness is actually required for something. Those who prefer to include Hausdorffness in the definition of compactness seem to be mostly influenced either by Bourbaki or by category theory.

There is no right or wrong in these cases. I believe that Bourbaki has “quasi compact” for the non Hausdorff case, but the terminology can be seen elsewhere.

The book where I learned topology is Kelley’s, where the Hausdorff property has its importance, but is not assumed throughout. Perhaps Engelking has his reasons for including the Hausdorff property when talking about compact spaces: indeed, it simplifies certain results because a compact subspace of a Hausdorff space is closed.

The prime example of a compact non Hausdorff space is any infinite set with the cofinite topology. Such a space is obviously compact, because the non empty open subsets have finite complement, so any open cover admits a finite subcover.

The simplest proof I remember of the equivalence between Zorn's lemma and Tychonov's theorem uses the cofinite topology, so non Hausdorff compact spaces do have their relevance.

Other important non Hausdorff compact spaces arise in algebraic geometry, when the Zariski topology is considered.