Another option is to use the Cut Time symbol -- a C with a vertical bar -- which is equivalent to 2/2 time signature. C without the vertical bar is known as "common time" aka 4/4 time signature.
When you indicate Cut Time or double time, you need not change the note values, it is understood that now the notes (and measures) following are all half the duration. So you could use your first example, with the Cut Time symbol, and can omit the metronome marking.
Note, sometimes composers use the doppio movement marking (mentioned in the other answer) in addition to the Cut Time symbol for extra clarity.
More info here:
Edit: This applies when you make a change from common time (4/4) to cut time (2/2). If you start the piece in 2/2, then of course you're simply starting it with a feeling of two beats per measure, strong beat every other beat.