I would argue that having double standards is not the same as taking context into consideration, as double standards implies that a different set of rules applies to a group of people based on a factor that has no logical or rational link to the rules you are enforcing.

Take, for example, choosing not to convict someone that is guilty of murder. Not convicting them because they have paranoid schizophrenia is taking context into consideration, while not convicting them because they are a woman is a double standard. That's because it's logical and rational to say that someone with schizophrenia might not be fully responsible for their actions, therefore may not be fully in control and responsible for the murder. However, most would argue that gender has no impact on one's judgment and propensity to kill, thus to not convict a woman just because she's a woman would be to apply these rules in an unfair and irrational manner, i.e. a double standard.

I guess in a way, I'm agreeing with your second point that not all standards are okay to double. However, that mainly comes from how the double standards that are okay have a sound reason behind them, and therefore we wouldn't consider them double standards in the way we usually understand the term "double standards".