Завод in Russian is also завод. Factory we usually translate as фабрика (Russian also has this word and spelling is the same, but pronunciation is different).
Basically, завод is where machinery and such is produced, фабрика -light industry. Food is a strange thing as some types of food are produced на заводі, some на фабриці
You are right! 'Factory' is the preferable translation for 'завод', not 'plant' which could make it confusing.