When someone tells you they want a text if you are going to be even one minute late, they are saying DO NOT BE LATE. Take an earlier bus, so that even if it gets stuck in traffic or you have to get off and do something then get on the next bus, you'll still be on time. Most days, you'll be ten or fifteen minutes early. The person who has given you these instructions is saying "do not time your arrival so you get here right on time; time it so you're normally early and will not be late." They are saying that your time is not as important as the group's time, and that you can do something useful when you arrive early, but the group can't do anything while waiting for an unknown amount of time.
You seem to worry that if you text "the traffic is really heavy today and my bus seems to be running 15 minutes late; I hope we make up the time later on the route but if we don't I only allowed 10 minutes delay in my plans so might be 5 minutes late" that this somehow leaves a "paper trail" that will be held against you. It will not. If you text, 1 minute before you should be arriving "looks like I'm 20 minutes behind schedule" that might be held against you, because it shows a lack of foresight or planning. But "I might be late" and then you weren't? Not a problem. (If you're worried the instructor is a nasty person who would use your 'might be late" texts against you even if you were not late in the end, simply send a "phew! made it with a minute to spare!" text when you arrive on time or early after sending a "might be late" text. But I doubt that is a serious worry.)
You may find this employer is not such a stickler once the training is complete. Or there may be an all-hands meeting every morning you can't be late for. Get used to planning for an early arrival (and having something to do for 10 or 15 minutes most days) and you will almost never be late.