Also see my answer to your other question.

EDS refers to energy-dispersive spectroscopy. The E does not refer to electrons. Several vendors sell XRF systems with EDS instead of WDS (wavelength-dispersive).

EDS collects in parallel and cheaper but is limited in count rate and resolution. That makes the detection limit poorer. For the SEM, I quote a detection limit around 0.1 wt%, but that depends A LOT on your setup.

EDS is often used with an SEM beam for excitation. That permits microanalysis. XRF always(?) uses an x-ray beam for excitation. That virtually eliminates the background and improves detection to the ppm range. Unfortunately, it is not a micro technique.

The term EDX is used to identify two different measurement techniques. One associated with electron microscopes (where an electron beam is the excitation source) and one associated with Xray Fluorescence instruments, where an Xray source is used for excitation.

(a) In scanning electron microscopy (SEM) and other associated electron microscopic techniques, the interaction of electrons give rise to characteristic X rays from elements. Therefore SEM instruments in general have an X-ray detection system attached to make use of this phenomena.

https://nano.oxinst.com/campaigns/what-is-eds/edx

(b) In XRF the source is usually an X-ray generator. Either an Xray tube or in some cases a radioactive source. This again gets classified as either wavelength dispersive (WD) or energy dispersive (ED). Here the difference is in the detection method. WD systems use crystals and detectors mounted on a goniometer, to disperse the xrays coming out of the sample (similar to creating an optical spectrum) and measure the signal at discreet areas. In EDX system however, the all emitted xrays are read by a single detector (usually a silicon drift type) and a spectrum is generated based on energy levels of the signals through electronic sorting.

There are obvious differences in terms of specificity and sensitivity between these two modes of detection.