The point of CIA is not to categorize a problem as fitting in just one of the three categories. The point of CIA is to mention desirable characteristics, and prioritize them. (Remembering the priority may be easy for people who have heard the acronym for America's Central Intelligence Agency.)
C = Confidentiality (is it private?)
Having a temporary problem with confidentiality can be worse than temporary lacking in the other areas, because this can be unrecoverable. Once data is leaked, then there may be no way to make it secret again.
I = Integrity (is it right?)
Is the data right? This is considered more important than Availability, because information needs to be right. If information is available, but the information is wrong, then people might expensively act on the wrong information. In contrast, if information is known to be unavailable, people might wait for right information, which is frequently better than spending resources making something happen (when it's not the right thing to happen). e.g., it's better to wait for directions, then to spend money/gas/time/etc. going East if you need to go West.
A = Availability (can I access it when I should be able to?)
Good to have. (For reasons explained above, temporary lapses in availability may be less painful than temporary lapses in the other areas. Still, people like to have access when they want it.)
Now, whether you agree with that ordering or not is perhaps less important. I've heard people give fine-sounding arguments why a different order might make more sense. Well, I say that if you're learning this for some certification/exam/etc., it is good to be familiar with the widely agreed-on official order (even if you're personally unconvinced that's actually best.
One reason we refer to this triad is that having the entire triad is crucial, and can be much more challenging than maintaining either of the other two.
- Making data public, but still read-only, compromises confidentiality while integrity and availability may be in tact.
- A faulty device driver leading to writing a blank sector, instead of desired data, might still honor confidentiality and provide (wrong) data upon request, indicating that there is still availability of data (even if it isn't the right data)
- Deleting data can help to really ensure confidentiality, and not violate integrity, but remove availability.
All three are useful/necessary.
I associate integrity to be concerned with safeguarding data from unauthorized modification, such as through MITM
Integrity has to do with rightness. An artificial MITM that changes data will result in lacking integrity. A flaw in the design of memory chips (e.g., "row hammer" attack) can result in a lack of integrity. A physical defect in media, which results in a bit being misread, can result in a lack of integrity.
"availability with safeguarding data from DoS such as through smurf or teardrop attacks."
Not just artificial malicious attacks. If a RAID array is offline (and not just degraded) due to physical defects, then you're lacking availability. If you have backups, you might not be suffering a loss of integrity (you aren't giving out bad information), but you lack availability until you get the drives into an online state.
I am inclined to say that unauthorized deletion of data is an attack on Availability principle given the data can no longer be accessed by legitimate users.
Example one: If I take a graphic file, such as a bar chart, and overwrite some of the pixels with white pixels (effectively "deleting" part of the bars), I violated integrity. If you can still open the file, and not realize there was a problem, then the data file is still available, but it is wrong. People looking at the data will think something false; when people find out you gave them false data, they may have grounds to consider you to be a spreader of false information. People may think you're just being dishonest. This situation is a problem with integrity, not availability.
Example two: If a log file removes content about an attack, the log file may indicate everything's fine. If you report everything's fine, you're saying something wrong. This ends up placing your reputation of integrity on the line.
Now, let's look at a case of availability:
Example One: If the file with the bar charts was deleted, then you can't open the file, so there is a problem with availability. Once the file is restored, your availability is fixed. At no time did you give out wrong data. Your trustworthiness is in tact; you maintained integrity.
Example Two: Similarly, when the log file is gone, you basically know there's a problem when you see "File not found". You're not being misled, so people aren't going to fault you for information that is misleading. This is not a problem with integrity; this is a problem with availability.
Is the unauthorized deletion of data considered a breach of integrity or availability?
I've just demonstrated how this could go either way, depending on whether the deleted data was just contents within a file, or an entire file missing. However, I'd like to re-iterate my earlier point: you could experience both problems at once. So don't feel like you have to categorize a potential problem as applying to just one category or the other. (It can be both.)
Edit (clarifying that a bit) - An attacker that steals information (violating confidentiality), defaces your web site's main page (so your web server provides undesired information, a lack of data integrity) and deletes the rest of your website including your backups (so that you can't get to the right information) results in a loss of the entire CIA triad. A single attack (whether one action, or more actions) can have multiple effects, and appropriately apply to multiple categories of the CIA triad.