You are correct, Richardson extrapolation applied to integration in general is the Romberg method. Perhaps some people like to make a distinction if not the full table with variable highest order is computed and used for the integral estimate.
For instance, applying Richardson extrapolation to the trapezoidal quadrature method gives Simpson's method. To apply R.ex. to the midpoint method one should use trisection of the segments to limit the growth of subdivision growth. This can be continued to get higher-order fixed-step methods.
You could also apply the error estimate of R.ex. to selectively decide which segment to subdivide to satisfy the desired error density. This would also be different from the Romberg integration.