It isn't about whether the load moves. It's about whether inertia — mass times acceleration — is large enough, relative to the elastic and damping forces, that ignoring it would give you the wrong answer.
It's tempting to draw the line at "does the load change over time" — a slowly ramped-up load feels static, a sudden impact feels dynamic. That instinct is close but not precise enough to be useful. A load can change over time and still be run as a static analysis, applied as a series of quasi-static snapshots, provided it changes slowly enough that the structure's mass never meaningfully accelerates in response. The actual test a structural analyst applies is quantitative: compare the frequency content of the applied load against the structure's own natural frequencies. If the load changes far slower than the structure could naturally respond, inertia is negligible and a static (or quasi-static) analysis is valid. If the load changes fast enough to excite the structure's own modes, inertia dominates, and only a dynamic analysis captures what actually happens.