

That’s correct as far as the TNG-era scale is concerned. In the TOS/SNW era it was a simple speed = warp factor3 equation, meaning Warp 6.25 is about 244c.
While not stated explicitly on screen, it was clear in behind-the-scenes documentation, and it was also clear that Enterprise in TOS exceeded Warp 10 in a handful of episodes, which I cited in my original comment. How fast a particular Warp Factor is may have been inconsistent, but the scale itself definitely changed between the two eras.
It was in the TOS Writer’s Guide as far back as April 17, 1967, where it was stated (page 8):
It was subsequently mentioned in the behind-the-scenes book The Making of Star Trek in 1968 and Franz Joseph’s Star Fleet Technical Manual. The TOS scale was finally made canonical when it appeared in on a viewscreen in ENT: “First Flight”.
The TNG scale was established in the series’ Writer’s Guide in 1987 establishing Warp 10 as the absolute limit (and infinite speed), so the scale had to be adjusted accordingly.