Further digging today seems to concur with CuriousOne's assessment. Short version: shortwave-or-solar radiation corresponds to irradiance integrated over 0.3-3 micron wavelengths.
From FAO's crop evapotranspiration paper: "Because the sun emits energy by means of electromagnetic waves characterized by short wavelengths, solar radiation is also referred to as shortwave radiation." This reference, however, still does not provide a quantitative indication of what "shortwave" means to a meteorologist.
Eppley Laboratory has a short primer on the measurement of solar radiation which clarifies: "Approximately 99% of solar, or shortwave, radiation at the earth's surface is contained in the region from 0.3 to 3.0 µm while most of terrestrial, or longwave, radiation is contained in the region from 3.5 to 50 µm."
One further twist: the instrument I mentioned in the original question is a LI-200SA pyranometer, which is what has been deployed in the Oklahoma MESONET. As it turns out, the calibration of this device is designed to produce measurements of "solar radiation" (0.3 to 3.0 µm) by assuming a fixed relationship between spectral response of the detector and the spectral shape of the incident irradiance (e.g., in clear, unobstructed daylight conditions). It does not give calibrated measurements of irradiance in the detector's 400-1100nm bandpass, as I initially thought.
So, the instrument gives calibrated measurements of solar irradiance integrated over the ~0.3-~3.0 micron band, which should be substantially similar to corresponding estimates of shortwave-or-solar "radiation" from climate models.