In my experience, the phrase in the best sense of the term is used to convey the positive opinion of the person using the phrase, or to describe someone or something in a positive light or manner.
To understand the meaning of this phrase, first you must understand that:
More often than not, words or terms have more than one connotation.
A word may have postive or negative connotation depending on the specific definition of the word. Often, people use context to clarify the intended meaning. For example, the term dark can have a negative connotation if you are talking about the contents of your day.
It was a dark day, and one that would be forever remembered.
Yet, it is also often used to describe men that have qualities or features thought to be appealing in males; in this context, it has a positive connotation.
He was tall, dark, and handsome.
Certain words, even with context, are hard to convey the intended meaning with.
Meaning the author wants to convey a certain connotation but the word either:
- has a connotation that is neither good nor bad - basically, a 'middle-ground' sort of feeling associated with it, OR
- has the opposite connotation (from the one the author wants to convey) in most contexts.
For example, the term plain - it isn't a word that is often used to describe good things. A lot of the time, you see it used like this:
The house was plain, and crudely built.
It is not often used as a positive descriptor (even when being plain is a good thing) due to the fact that it has taken on a negative connotation in most settings.
Now, what if a person found themselves in one of the above two situations? They have decided that some particular word is essential in order to give a proper positive opinion, or positive description of something or someone.
Yet the term has many different definitions with different connotations, or has mostly negative connotations.
This is the situation in which they would use the phrase "in the best sense of the term". It is because they are trying to convey that the intended meaning of the term in this situation is the most positive one possible.
She was plain, in the best sense of the word. Nothing extravagant, nothing excess, yet not lacking in character either. Comfortable with herself, and I was comfortable with that.
In very generalized terms, it means:
If you meet the stated requirements, I will let you proceed. If you clearly do not meet the stated requirements and are attempting to convince me that I should let you proceed anyway, I will force you to leave.
Most commonly, the "stated requirements" are "can you pay the amount I am demanding", i.e. "do you have enough money". For a literal application of the phrase, consider a couple of poor college students trying to get into a party for which there is a $25 entrance fee. If they show up without enough money, and instead of paying they attempt to convince the gatekeeper that they are friends of the band, he might well tell them "Money talks, BS walks" as he refuses them entry.
Best Answer
No, it doesn't mean that (at least not in the US).
As a native speaker in the US, if I heard someone say "hit and trial" I would probably stop them and ask if they meant "hit and miss" or "trial and error".
I've never heard that phrasing until seeing this question.
In your example question from Mathematics SE, it looks like they mean "trial and error", but I get that from the context, not from understanding the phrase.
It might mean that in Indian English.
It's been suggested that this might be a phrase that's used in Indian English.
I can neither confirm nor deny this, but a google search reveals it being used to mean "trial and error" in an article written by a "Brajesh Shukla", in a question asked by a "Surya Varma", and in an article on "The Hindu". So, the suggestion seems as though it may be true.