Kindergarten to Americans: when most children start elementary school, usually age 5 (maybe 6 if their parents didn’t think they were quite ready), Grade 0. What Germans would call “Vorschuljahr” (last year of Kindergarten, where there’s less play and more learning how to count, recognize letters, and sit still at a table)
Kindergarten to Germans: usually its own establishment, available but not mandatory from age 3, usually has fees in the 200 EUR/mo range for all day with hot lunch (with further subsidies for poorer families), rather like an American preschool for ages 3-5, with the 5-6 year (“Vorschuljahr”) being more like American kindergarten.
The Kindergarten to Germans you're describing seems like the "Kita": https://de.wikipedia.org/wiki/Kindertagesst%C3%A4tte - but the whole education system is very confusing and it's generally not language-specific but more culturally specific AFAIK
Kindergarten to Germans: usually its own establishment, available but not mandatory from age 3, usually has fees in the 200 EUR/mo range for all day with hot lunch (with further subsidies for poorer families), rather like an American preschool for ages 3-5, with the 5-6 year (“Vorschuljahr”) being more like American kindergarten.