It might be worth noting that the word "Feld" is also (sometimes) used in German, at least in the compound "Galoisfeld" (for "Galois field"). And, this is not some modern-day re-translation from English but classical usage (of Witt for example), see the discussion in comments of http://mathoverflow.net/a/18638/9072 for references.

In my opinion this provides additional support for anon's answer that quite at the start there were two different names and just in the one language the one 'won' whereas in the other one the other 'won' the 'competition'.

Regarding other theories and in general, it might be worth noting that "Körper" does not only mean body, like in human-body, but is also used to designate certain structures/organizations/groups, like "Justizkörper" (comprising judges, attorneys, and so on). And, in that meaning "Körper" fits a lot better with "Gruppe" (group) and also with "Ring" (also used as name for organizations, like in English) and also "Verband" (again used to designate organizations, the math meaning being lattice, in the order sense, providing another example where it is not a translation). See http://mathoverflow.net/a/117354/9072 for some discussion. So that "Körper" is really not all that sensual in that context but quite corporate (which ultimately derives from the same source, I think).

It might be worth noting that the word "Feld" is also (sometimes) used in German, at least in the compound "Galoisfeld" (for "Galois field"). And, this is not some modern-day re-translation from English but classical usage (of Witt for example), see the discussion in comments of https://mathoverflow.net/a/18638/9072 for references.

In my opinion this provides additional support for anon's answer that quite at the start there were two different names and just in the one language the one 'won' whereas in the other one the other 'won' the 'competition'.

Regarding other theories and in general, it might be worth noting that "Körper" does not only mean body, like in human-body, but is also used to designate certain structures/organizations/groups, like "Justizkörper" (comprising judges, attorneys, and so on). And, in that meaning "Körper" fits a lot better with "Gruppe" (group) and also with "Ring" (also used as name for organizations, like in English) and also "Verband" (again used to designate organizations, the math meaning being lattice, in the order sense, providing another example where it is not a translation). See https://mathoverflow.net/a/117354/9072 for some discussion. So that "Körper" is really not all that sensual in that context but quite corporate (which ultimately derives from the same source, I think).