What's the bias you see in the statements "King Louis was executed" or "former king Louis executed by means of guillotine for high treason"? Both the terms "execution" and "high treason" are objective descriptions takens from the proceedings.
So no, you can't have it both ways. You can't point to objective reporting and claim it has a bias, when in reality you mean that the reporting isn't showing your preferred bias.
Both are true, and which term you use just depends on your point of view. Both are true, but not unbiased.
There is no proper objectivity outside of the sciences. Everything in culture, society and politics depends on an everchanging inconsistent and observer-dependent frame of reference. You can be kind-of objective if all of you and your local neighbourhood agree on a common frame of reference. But you always have to be aware of the lack of global objectivity.
Critics agree that Mr Obama's middle name is Hussein and he is likely an islamic terrorist.
The first part is obviously true. For the second part, I am fairly certain I can find two people on 4chan who agrees with this, and they will certainly be critics of Obama.
Statements of objective truths can still be biased.