Steam-distilled water will have some trace minerals in it due to its loosely-controlled commercial process, but it won't damage your steam iron or lead-acid battery if you use it to top them off.
De-ionized water is almost completely free of mineral contamination, but is more aggressive corrosion-wise because the missing calcium and magnesium salts are what buffer the water's pH to a near neutral 7. The de-ionization process requires two resin exchanges, one to exchange the cations (Ca & Mg for instance) for H+ (hydronium) ions and the second to exchange the anions for OH- (hydroxide) ions. during the process, the lonely H2 and OH ions instantly combine to form pure H2O (hydrogen hydroxide or pure water, something you should not consume in mass quantities).
Plain ion-exchanged water is the same thing as softened water. After softening, the only cations it contains are sodium (Na+). They are in the form of salts such as sodium carbonate converted from calcium carbonate (lime or CaCO3) for instance. In this example, the sodium carbonate is actually washing soda, a common laundry additive that makes soap "soapier" and prevents the formation of soap scum (calcium stearate). You may be disappointed to learn that calcium stearate is used as a food and cosmetic additive to provide thickening, but as they say, "waste not, want not".
Steam-distilled and de-ionized water taste oddly terrible because of the virtually total absence of taste-imparting trace minerals.
Water is one of the most complex simple compounds you'll encounter in everyday life. I hope this clarifies a few points. All of this information can be found online.