All engines consume oil, but its usually in the very low amount between services.
if oil is breaking down due to age, use etc, then it will be burnt more readily, and you will notice it more.
first things first, give it a service, assuming correct grade oil is used, and the correct amount.
Watch its level for the first 1k miles, then the next, you may notice its oil level stays pretty much spot on until you get up with the miles and time, then you may notice it drops quicker, and or more the next time you check.
Just because your oil level is dropping, its not always a sign the engine is failing, check the basics first, then if your getting high oil use with fresh oil, and in the first 1k miles, then go further.
anyway, thats all fluff, you need to find out why the warning lights came on, worry about the oil burn later.
yeah, sounds like short cold trips, thats hard on oil.
I have an old diesel hilux 2.8, a very sooty motor being indirect injection, and I change out the oil every 2500-3200kms since it mainly does short trips in and out of town and for firewood runs.
If you ignore keeping the servicing up to those motors, they soot up, and the oil thickens, and eventually you clog oil galleries and they eat bearings.
keep the oil clean (within good servicing paramaters, ie change out once its saturated and cant hold solids in suspention anymore), and they last for quite some time.
edited to add
Petrol motors also affect oil, but its more towards dilution due to the petrol, sort of an opposite effect than diesel, but it shows that oil has a serviceable life, its job is to protect against wear, but also to carry wear particles in suspention for it to be then filtered out when it passes the oil filter.
oils have additives to make them suitable for a select situation, and depending upon the way the engine is used, and its condition to start with, will determine oil life.
I have used diesel oils in petrol cars that had quite a bit of carbon and oxidised oil that left lots of deposits inside the motor, including jamming up the oil control rings etc.
running the high detergent diesel oil in the petrol motor enabled the diesel oil to clean away those deposits over time, and bring back the motor from high oil usage and high blow by in the rings to usable for many years more. I dont like the idea of an oil change engine flush add to motor and run before draining the oil, better to let the detergents in the diesel oil do the job more slowly and better, less chance of clogging up oil galleries.
doing that works well on the subaru boxer engines that have issues with oil sticking the piston rings.