The two sections you quote are not directly comparable, I think. Some background: In OLS regression, the assumptions of the process (which are not always considered by its users, as noted by the author of that article) include the requirement that the independent variable(s) be measured perfectly--that is, with zero measurement error. This condition, like many in formal statistical models, is impossible to meet in any many real data situations (not _any_... thanks, glen_b). However--like with many statistical problems--failing to meet this condition is often considered (by the experts) to have little negative consequence for data analysis. In reality, the measurement error of both the IVs and the DV are probably represented in the overall "badness of fit" of the model (the "residuals" and their many associated statistics).
Despite the above, I think the author is saying that failing to account for the reality of measurement error in all variables (i.e., IVs) can be problematic in many situations. I assume this is why there is a family of models that include estimation of measurement error in the IVs. I think structural equation modeling is like this, too, as most variables in SEM have error terms associated with them.
Finally trying to directly answer your question: The first quote is saying that in OLS regression we assume the IV is measured with no measurement error, which is clearly impossible, but (I think it's saying) that is OK when using OLS for prediction. The second quote seems to be something different: part of the description of the tasks necessary for functional analysis (probably, since that's what the article is about). In the second quote, the author is explaining the difficulty of modeling when the residual (i.e., error) variance in a model must be accurately identified as either coming from the IV or from the DV, instead of (as in OLS) just assuming it all comes from the DV. This is not directly relevant to the first quote in the context of your question.
...both methods should be used when both variables are in error.
All measured variables always have measurement error, so that statement might be based on your misunderstanding of some issues. Rather, the author seems to be saying (in the first quote) exactly what is in the article's abstract: that OLS is not appropriate in earth science except in prediction situations. The second quote is not directly related to your question, as it is a description of a part of the alternative method that the author favors.