Self-refuting statement?

The statement "there is no objective truth" (type E) is well-known to be self-refuting, because in order for it to be true, it has to be false.

If the statement is true, then it is objective that there is no objective truth (as in, it isn't subject to artefacts like cultural bias). However, that makes the claim that there is no such thing as objectivity false.

If the statement is false, then there is some sort of objective truth that exists...so the claim that there is no objectivity is also false.

So, universal negatives can be self-refuting. But what about universal affirmatives?

Consider the statement, "every principle has exceptions" (type A). That's elf-refuting too, isn't it? No?

If the statement is true, then the principle that there is an exception to every principle implies that this principle itself has exceptions, which implies that not every statement has exceptions. The statement cannot be true.

But if it is false, then the statement "every principle has exceptions" does not have exceptions - but this contradicts the idea that "every" single one of them does. The statement cannot be false.

This seems asymmetrical with the first example. In the first one, the statement's truth is perplexing (but not its falsity). In the second one, they're both perplexing (because truth leads to falsity, and falsity leads to truth, creating a paradox).

The first one is contradictio in adjecto, but is the second also an example of this fallacy, or a logic problem?

Always start at the beginning and get the definitions and concepts agreed to, they are, after all, such slippery things. 

I think we can agree that we can never be “absolutely” certain of anything other than metaphysical axioms, so we must assign value to any proposition on the available evidence.

For our discussion here, I’ll say: “An axiom is a proposition that defeats its opponents because  they have to use it in any attempt to deny it.”

We can, however, attain an epistemological certainty, which, loosely restated, means certainty beyond a reasonable doubt, while keeping our minds open to additional evidence or a different interpretation of available evidence.

The most fundamental, irreducible and absolute axiom states: existence exists. Any attempt to deny these two words requires the acceptance of the existence of the argument used in the refutation, and, therefore, falls of its own weight. And, because of this, it is also called the Fallacy of the Stolen Concept. In the statement under analysis, there is also a bit of the Fallacy of Equivocation. 

I am having a problem getting past:

The statement "there is no objective truth" (type E) is well-known to be self-refuting, because in order for it to be true, it has to be false.

Those who don't believe in objective truth, are not claiming anything is objectively true. It would be self-refuting by objective truth standards but not by the standards by which one who claims this abides. I think I would need to know what the person saying means precisely by "objective truth" to see if this MUST be self-refuting or not. I apologize if this is commonly accepted idea in philosophy that I am just missing.