Hello again! The "shi" in bushido means "warrior" and refers to the warrior who serves a liege lord and embodies the (Confucian) virtues of loyalty, duty, honor, courage, filial piety, compassion, etc. . . . . Bushido is not an old term but its first well-known usage was coined in the classic, Hagakure Bushido, a collection of didactic stories written by lords of the Nabeshima clan for their samurai retainers. Budo refers to martial arts - generally modern martial arts, - Aikido, Judo, Kendo, Karate-do, etc. but through usage has now to come include all types of martial arts. Bushido and Budo are really not exactly interchangable terms. As you say, Bushido refers more to the ethical and moral code of the warrior. Budo refers to martial arts or martial science.
Because the term "bushido" was often used as a propaganda tool by the army during WWII, "bushido" to many Japanese today has a nuance of ultra right-wing conservatism and is not often used. Mishima Yukio, the noted author who killed himself at the Self Defense Hdqtrs several decades ago, also wrote his own version called, "The Modern Hagakure Bushido."
"Shi" meaning warrior has a connotation of the Confucian ideal of "chun-tze" or the scholar-gentlemen - the ideal of society and model of propriety and behavior.
Because Japan was a feudal nation for over 800 years, there are numerous terms referring to the warrior. . . . bushi, heihyosha, samurai, shi, etc. Hope this helps a little. Thanks!