Cusp
A cusp is the boundary line of a house, the degree at which one of the twelve divisions begins, and by extension the point where a sign gives way to the next.
House cusps are what a house system computes, and they are why two systems can disagree about which house a planet is in while agreeing perfectly about where the planet is. The planet's position is astronomy; the cusp is a convention.
The popular sense, being born on the cusp of two signs, is a different use of the word and a looser one. A planet is in one sign or the other with no overlap, though a body in the last degree of a sign is genuinely about to change its whole context.
Frequently asked
Can I be born on the cusp of two signs?
Not in the technical sense. A planet occupies one sign only, though being in its final degree does put it on the edge of a change.
Why do house cusps move between systems?
Because each system divides space or time differently. The planets stay put; the boundaries drawn around them do not.