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I swear that exactly half of the many, many sites I found say that our Galaxy and its immediate neighbors are moving towards the Leo Constellation (I believe) and, ultimately, supposedly, the 'Great Attractor' at about 1.3 million mph or about 370 mps) and the other half claim it is moving at about 830,000 mph (about 370 kps).

Does anybody have a definitive answer?

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    $\begingroup$ The center of the Milky Way moves at 565±5 km/s (as answered here). In Planck Collab. et al. 2020, you can see how the MW center, the Local Group, the Local Standard of Rest, and the Solar System move through the Universe (i.e. wrt. the CMB). $\endgroup$
    – pela
    Commented Dec 29, 2021 at 10:46

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The two values are the peculiar velocity of the local group relative to the Cosmic microwave background

The Local Group — the galaxy group that includes our own Milky Way galaxy — appears to be moving at 627±22 km/s [390±14 mps ] in the direction of galactic longitude ℓ = 276°±3°, b = 30°±3°

Or the peculiar velocity of the solar system relative to the CMB:

The Sun appears to be moving at 368±2 km/s relative to the reference frame of the CMB (also called the CMB rest frame, or the frame of reference in which there is no motion through the CMB).

(wikipeida citing Scott and Smoot)

The values are different because the sun is also moving in an orbit in the Milky Way, and the Milky way itself has a peculiar motion relative to the centre of mass of the local group.

It is just a coincidence that the value in km/s of the latter is close to to the value in mps of the former.

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