Although the vertebral column is segmented into vertebral bodies and discs, true 'segments' probably don't exist in the spinal cord. Rootlets don't emerge from the spinal cord in bundles according to vertebral level (although, they do exit the vertebral column foramina in just such bundles as nerve roots), instead rootlets line the dorsal and ventral horns of the spinal cord in an un-segmented strip with an appearance much like the insertion of nylon hair on a plastic dolly's head (as beautifully demonstrated in this 2016 paper by Leijnse & D'Herde, https://doi.org/10.1111/joa.12493)
Despite these caveats, grand efforts have been made to work out the probabilistic correspondence between vertebral level and spinal cord level, an important task for those interested in attributing function or pathology to certain cord segments in vivo. Cadotte et al. 2015 (https://doi.org/10.3174/ajnr.A4192) have produced a wonderful paper with this aim in mind which includes some useful probabilistic maps (which, for those interested are integrated as part of the Spinal Cord Toolbox: https://sourceforge.net/p/spinalcordtoolbox/wiki/Spinal_levels/)
Long answer ...
Answer part 1: Cord 'segments' (defined by the approximate length of cord from which rootlets forming a single spinal root) emerge by definition with an offset from the vertebra they are named after (see Part 2). Furthermore, the offset between rootlet exit from the cord and root exit from the vertebral foramina (i) increases the further down the cord you look and (ii) varies amongst individuals.
Answer part 2: The naming convention of vertebral bodies and roots entails an offset between vertebral bodies and their root namesakes: cervical (neck) roots are always numbered according to the vertebra above which they emerges, e.g. the C2 root emerges between C1 and C2 and thus the C2 rootlets (and by implication the C2 cord 'segment') are likely, at least in part, to be in line with the C1 vertebra:
By convention, however, the root that emerges above the first thoracic (back) vertebra is named 'C8', and from anywhere below that level, roots are named according to the vertebra they emerge below:
This step change in nomenclature means that for a few of the upper thoracic verterbra, the cord rootlet exit zone for a numbered root is anatomically in line with the synonymous verterbra, i.e. the rootlets that merge to form the T1 root exit the cord at the same level as the T1 vertebra. (NB this is not shown on the figures, but the Spinal Cord Toolbox link above has a probability hotspot diagram that demonstrates this nicely).
