A trilogy - if it really is conceived as a trilogy from the start - or Shakespeare's "two-parters" aren't the same thing as sequels, in the modern definition (although that are sequential). The Henry IV plays are modelled on the old medieval biblical cycles, and are this conceived as a single work in multiple parts; properly speaking, sequels and prequels are later texts grafted onto an original.
But a single story told in three parts is not a trilogy by the proper definition. Formally, a trilogy is three
separate but connected works.
The Lord of the Rings is not technically a trilogy since it was meant as a single story and was published in three volumes for reasons of length. The Raimi
Spider-Man films, say, are a trilogy, because each one is a separate story but they collectively form a larger arc.
So I don't agree that "original design" has anything to do with it. What you're saying only applies if it's a
single story broken into pieces, but that's not the only kind of original design. One can certainly have the design to tell multiple distinct but connected stories, and it's perfectly valid to call those sequels.
So no, technically
Henry IV Part 2 isn't a sequel to Part 1, but Part 1 is a sequel to
Richard II and
Henry V is a sequel to
IV.
Before Watchmen is a prequel to Watchmen because it was not part of the original design.
No, it's a prequel only because it's not part of the
same single work, which is a different thing. More basically, it's a "prequel" because it's a sequel (a work coming out later) that's set before the thing it's a sequel to. The term is basically a portmanteau for "preceding sequel."
Empire Strikes Back is not a sequel because it was part of the original design.
Both parts of that are wrong. First, it
is a sequel because it's a distinct story; the previous film didn't end on a cliffhanger or in the middle of the particular tale it was telling. The two stories were parts of a larger arc, but each one had its own distinct arc. Second, it wasn't part of the original design. Lucas has grossly rewritten history in his claims about how much he had planned in advance. At the time he wrote and made the original film, he had no idea that Darth Vader would turn out to be Luke's father. Hell, he had no idea he'd even get to do a second movie at all, or that he'd have the budget to make anything remotely near as ambitious as TESB. He had Alan Dean Foster write the novel
Splinter of the Mind's Eye as the basis for a potential low-budget sequel, and it told a completely different story from TESB. It just doesn't make sense to talk about the original
Star Wars trilogy as a single unified work. The prequels can be thought of in that way, but it's definitely untrue of the originals.