Let me break these into groups, because the English adjectives you list are not synonyms. Hopefully you'll get a sense from this that there isn't so much of a broad term for people and cultures that may share a culture and language, but ultimately are every bit as distinct as the USA, Australia, and South Africa are.
"Chicano" specifically refers to Mexican Americans, and has strong political associations.
Hispanic, Latin, Latino
These all ultimately refer to the Roman Empire, which expanded into the Iberian peninsula fairly early in its history. The Romans referred to the province as Hispania. Note that relating back to that peninsula can have political meaning, and does not necessarily imply a difference between Spain and Portugal as the historical colonial power that colonized an area. If someone wanted to be more explicit about an ethnic tie back to Europe, they may use "criollo" (same root as the English "creole").
"Spanish-speaking" is probably most directly translated as "hispanohablante," but that again doesn't necessarily mean "que habla Español." Note also that in Spain, what you might think of as the Spanish language is generally referred to as "castellano" (Castilian Spanish), as there are many Spanish dialects—and again, these dialects are often tied to deep political and ethnic beliefs.