There's no difference in the calculations, but the Moon, Sun, planets don't have more or less fixed right ascensions and declinations like all the stars. The site you linked seems to only work for stars. Where are you getting ephemerides for non-stars?
WARNING: This is an attempt to answer. The reason is that I thought I understood the TEME concept until, trying to answer your question leads me to be more cautious. Especially, when I discovered that "TEME" may have different definitions, interpretations, in various implementations of SGP4 (cf ).
My answer will be based largely on this paper ...
Most of your assumptions are correct.
However, right ascension and declination are an equatorial system, while (heliocentric or geocentric) longitude and latitude are an ecliptic system. The reference planes do not coincide, even though the main reference point, the vernal equinox, does—this is because that point is where the ecliptic crosses the equator ...
FK5 is based on optical observations. ICRS is based on radio data. Radio observations (ironically) are much higher accuracy than optical through the application of long baseline interferometry.
Other than extremely accurate astrometry (fractions of an arcsec), they are essentially identical.