"ionized" as in marketing and DI/distilled have diddly squat to do with one another.
"ionized" water is marketing. the amount of dissociation Kw style is purely dependent on Kw itself and all other ion species present. The "ionizing" they speak of is submitting it to electrical (and therefore also magnetic fields). That effect goes away the second the field goes away...
Ph your water to where you like it the old way. Remove the species who's taste you don't dig on. I like hard water, so I am cool with that. I hate metallic tasting water (thats why this one doesn't like brewing boilers.), so I avoid that.
An ionizer won't make it alkaline or acidic. I promise. it will do nothing. It is not possible to change the combined dissociation equilibria in solution via current. Adding and removing species... that's all one can do. And "ionizers" don't do either.
Hope this didn't tick anyone off. I did not mean it like that.
Felix