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This interval is called the confidence interval, and the radius (half the interval) is called the margin of error, corresponding to a 95% confidence level. Generally, at a confidence level γ {\displaystyle \gamma } , a sample sized n {\displaystyle n} of a population having expected standard deviation σ {\displaystyle \sigma } has a margin of ...
So, I would argue, conflating the "margin of error" with the "standard error" or standard error-based statistics is to perpetuate confusion and misinterpretation for potentially high-stakes decisions (as examples in the article demonstrate).
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The problem with "margin of error" is that it sounds like it should be much more than it is. Really, all it is, like I noted in the opening, is a transformation of the sample size that tries to reflect sampling variance.
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The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled).