For badcontinue errors, the API currently returns a code of either "??badcontinue" or "??_badcontinue" (where ?? is the module's prefix), depending on the module. The human-readable error message text may also vary depending on the module.
In 1.21wmf8,[1] these will be standardized as "??badcontinue", with a human-readable message of "Invalid continue param. You should pass the original value returned by the previous query".
Since well-behaved API clients should never see this error message, this change should not require any client updates.
[1]: Gerrit change https://gerrit.wikimedia.org/r/#/c/42398