Is it correct to say that the word にち nichi /'ni.tʃɪ:/ means in the same time sun, day and Sunday?
Yes, the meaning was borrowed from the Western calendar when Japan adopted that calendar in 1872. Previously, Japan used a ten-day week.
Also the same are Monday (= moon day) and 月曜日, and Saturday (= Saturn's day) and 土曜日 (the planet Saturn is 土星).
The other four days in English come from Old English god names: Tuesday (Tiw's day), Wednesday (Oden's day), Thursday (Thor's day), and Friday (Frige's day).