Learn English – What Does “easy on the” Mean in Food Preparation

You catch more flies with honey than vinegar or, sometimes you catch more flies with honey is an English proverb. It doesn't have a counter-intuitive meaning--if you are trying to catch flies, you are literally going to attract more with honey. That is, you're going to get what you want (in the proverb flies, but in life any goal) with sweetness rather than acidity.

This answer explains it similarly:

Flies represents anything you want to achieve. Honey (sweet) represents anything pleasant that you do to get what you want. Vinegar (sour) represents anything unpleasant that you do to get what you want. It tells you to use nice methods rather than unkind methods in dealing with other people.

This is a saying that means: you will be more successful in life being sweeter, or nice rather than being, mean to people, not nice and doing hurtful, dishonest things in life.

This forum makes some guesses at its origins, noting:

The proverb has been traced back to G. Torriano's 'Common Place of Italian Proverbs' . It first appeared in the United States in Benjamin Franklin's 'Poor Richard's Almanac' in 1744, and is found in varying forms..." From "Random House Dictionary of Popular Proverbs and Sayings" by Gregory Y. Titelman (Random House, New York, 1996).

This use of the word "presidential" is using Oxford English Dictionary definition 1.b.

U.S. Having a bearing or demeanour befitting a president; dignified; confident. Also: appropriate to a president; stately; impressive.

The word was originally used to refer to presidential candidates, as opposed to a sitting president like President Trump, when discussing whether their behavior and conduct was of the kind that made them fit to be a president. That is still the way in which it is most commonly used.

During the 2016 U.S. presidential election, there was much discussion about whether candidate Trump had a demeanor that was presidential, or befitting a president.

When asked about his demeanor during the campaign, Trump famously said that if he was elected president:

"I will be so presidential, you will be so bored."

The writer in the Washington Post is insinuating that President Trump's remarks about Charlottesville were not appropriate for the office of President, or in other words not presidential. The writer is contrasting that behavior with what a presidential president would have said, by which they mean, these are the remarks that befit the office of President.

Edit: In your second question, you ask:

Do you say 'vice presidential vice-president, 'chairpersonal chairparson,'directorial director,' 'managerial manager, 'doctoral doctor,''paternal father, maternal mother,' 'childish child,' and likewise?

The reason most of these phrases would be unnatural is because the adjective presidential meaning "befitting a president" is unique. It is an outgrowth of usage.

As with most adjectives that us the -al or -ial suffix, its initial meaning was of or pertaining to the noun, as with OED 1.a.

Of or relating to a president or presidency.

So uses in this sense would be something like "The presidential campaign was underway." Uses like this were extended to the sense cited at the top of this answer gradually over a long period of time. You can see the difference between the meanings in this sentence:

X, who is a presidential [sense 1a] candidate, has an appearance that is very presidential [sense 1b].

When discussing who should be chosen for president, it was and is reasonable to ask, "Does this person have a presidential appearance/demeanor/style?"

However, directorial, managerial, and doctoral never underwent these shifts in usage.

The only sure way to know whether an adjective derived from a noun can be applied to itself this way is to consult the entry for the adjective in a thorough dictionary.