Why does a cyclist at speed push the right handlebar to start a right turn?

Answer: It creates the right lean

It points the wheel right — Wrong. Pushing the right bar does not point the wheel right; it steers the front wheel left for an instant, which moves the bike's support path left and makes the bike lean right. Only after that lean is established does the wheel steer into the right turn.

It creates the right lean ✓ — Correct. A bike must lean into a turn, and countersteering is the quickest way to create that lean. To turn right, you first steer left, which tips the bike right; then the wheel follows into the right turn.

It moves body weight — Wrong. Body position can affect riding, but this specific push works by changing steering and roll, not by simply shifting weight. The handlebar input first steers the bike the wrong way briefly, and that is what creates the lean.