Can I say "I slept in and was late for work" in this situation?
You can absolutely say that, sleeping past your intended time to wake up is "I slept in", were you late for work? If yes, then this is still a true statement. But it does leave the intent ambiguous compared to other phrases, which can often be a good thing.
Does "I slept in" imply I did it on purpose or by accident?
In most cases, the reason why you slept in or if the act was on purpose or an accident is irrelevant. It is a neutral term really with regard to your intent, you have described the action without giving any context. This means that the recipient of this information is going to process it with regard to how your tardiness has affected them, how late you were for the engagement and whether you arrive at all will further impact the outcome.
As a general phrase, "I slept in" is used because by not disclosing exactly why you slept in or if you did it on purpose or not means that you can rely on existing goodwill to help determine the outcome.
- In any scenario, if you were to sleep in often, then it demonstrates lack of effort and lack of regard or disrespect for the engagement that you missed.
Other phrases like "I had a lie-in today" or "I chose the sleep in" or "I turned my alarm off last night" clearly define that this act was on purpose and pre-meditated (it is reasonable to expect that those actions would lead to you sleeping in too long), which will generally reflect poorly on you, but will sometimes get you credit for honesty.
- In the context of work this is a common phrase to use. It is generally received as an excuse. The fact that you slept in deliberately because the you snoozed the alarm or slept through it or forgot to set it in the first place is irrelevant. To arrive at work at the prescribed time is your responsibility, so it is incumbent on you to ensure that necessary precautions are taken to get you there on time.
Even in scenarios where slept in due to power-loss or your dog ate your phone or alarm clock, your intent is irrelevant. The more specific or elaborate that your excuse might be to try and justify why you did not wake up is even more likely to reflect poorly on you, this is another reason why using the simple phrase "I slept in" is helpful, it avoids the suspicion that you are trying to cover up the fact that you slept in on purpose or had a reasonable expectation before you went to sleep that you would sleep in, even if you did not do it specifically on purpose.
The number one reason why we sleep is that our body did not get enough sleep in the first place. That might be due to a medical condition, the environment you sleep in, having kids or a partner, or pets that keep waking you up, sleeping with the TV on, staying up too late, drinking or taking other substances whether they be medicinal or not before going to bed...
It is generally accepted that All of these things are to a degree within your control. If it is important for you to wake up on time in the morning, then you will take whatever precaution that is needed to ensure that these other factors that affect you, will not prevent you from waking up on time.
If you were talking about the fact that you "slept in, and was late for work" to a third party who was unaffected by this, then you might choose to disclose why you think or why you deliberately slept in. But if you were talking to someone who was affected, either because it was important to them or your lateness/absence meant that they had to put in more effort to cover you then they will not be impressed to hear that you did it on purpose, nor would they be happy to hear that you didn't put in the necessary effort to ensure that you were there on time. For these people keeping it short with "I slept in" is generally the best defence you can offer, it is probably better not to disclose any further details. However, it would be polite to apologise :)
- The only generally acceptable excuse for lateness is medical or major environmental emergencies that are truly beyond your control. Every other excuse is simply that, and excuse and admission that you didn't put in the required effort.