What some people do and think they did it correctly isn’t an issue. You asked when shacharis starts. it starts when shulchan orech says it starts. Shacharis strts at brochos, or ctully at “mah tovu”. The oart of korbonos and kitores is part of shacharis even if some people skip it. it is still part of shacharis.
shacharis ends with oleinu for nusach sfard and at the yom for the day for aschkenaz. Deciding that you have no time doesn’t mean that you davened completely. These are the times of a complete schacharis. Period. Halacha.