Short answer: the four terms describe overlapping work on the same problem, and only two of the distinctions between them change any decision. Use GEO for whether a brand gets named, AEO for whether a page can supply the answer, and treat AI SEO and LLMO as umbrella labels.
The four terms
GEO, generative engine optimization, is about whether an assistant names, describes and cites a brand in the answers it writes. The unit of work is a buyer question.
AEO, answer engine optimization, is about whether a page yields a correct, self-contained answer that an engine can lift and attribute. The unit of work is a passage.
LLMO, large language model optimization, is the widest label. In principle it covers what a model absorbed in training as well as what it retrieves at answer time; in practice nearly all published LLMO advice is about the retrieval half, because that is the half anyone can affect this quarter.
AI SEO usually means classic SEO practice pointed at AI surfaces, often by teams whose existing tooling added an AI report. It is the least precise of the four and the most likely to describe a dashboard rather than a method.
The two distinctions that matter
The first is passage versus brand. Making a page extractable and getting a brand recommended are different projects with different owners. The first is editing and markup on your own domain and you can finish it. The second depends on what third parties say and you can only influence it.
The second is training versus retrieval. If an assistant only names you when web search is enabled, you exist in the retrieval half only, and this month's publishing can move the needle. If it names you with search off, you are in the model's absorbed knowledge, which is slow to change in either direction.
Everything else in the vocabulary is branding. These two distinctions decide who does the work and how fast it can possibly pay off.
What does not change
Crawlability, clear factual statements, structured data that matches the page, dated content, and a consistently described entity are prerequisites under all four names. If a vendor's method sounds entirely new, it is usually either those fundamentals renamed or something unverifiable.
Which term to use
Inside a team, name the thing you are measuring rather than the discipline. "Mention rate on our thirty tracked questions" survives a change of vocabulary; "our GEO score" does not. Externally, GEO currently carries the most search demand of the four, which is a reason to use it in public writing and not a reason to believe it is a different activity.
A practical division of labour
Give AEO to whoever owns the pages: headings that are questions, answers first, facts as sentences, markup that agrees with the page. Give GEO to whoever owns demand: which questions to track, which third-party sources the answers cite in your category, and what to publish or place there. Measure both against the same fixed question set so the two teams argue about evidence rather than about terminology.