What is the difference between an 'idea' and a 'concept'?
Has a distinction between the two terms been settled and accepted by most modern philosophers?
I don't think you can find a consistent distiction across philosophers.
According to Frege (german logician, end of 19th century), ideas are mental images (psychological objects), concepts are objective.
He speaks of "conceptual content" of a linguistic expression, meaning something that is communicable, i.e. intersubjective.
See Begriffschrift (1879) or Sense and Reference(Uber Sinn und Bedeutung, 1892).
Ideas is a term that goes back to Plato. For Plato, this is another word for the forms.
I am not sure where the use of the term concept originates in philosophy; it's at least in Aristotle and Kant in different ways.
I am not very familiar with Aristotle's usage, but you can read more at the SEP.
For Kant, the concepts are the fundamental categories of the understanding (the faculty through which we know). When he uses the term ideas he means thoughts we have formed about the world that help to frame other thoughts but do not form fundamental categories of the understanding.
For Kant, there are exactly 12 categories that the mind uses to organize reality that are in four groups at Critique of Pure Reason A80/B106:
Quantity: Unity, Plurality, Totality Quality: Reality, Negation, Limitation Relation: Inherence and Subsistence (substance and accident), Causality and Dependence (cause and effect), Community (reciprocity) Modality: Possibility, Existence, Necessity
Kant has an argument for why he thinks it's those 12.
Hegel inherits Kant's usage and defines the Concept (capitalized) as the overarching structure of reality. Thus for Hegel, the concept is not ultimately the 12 categories of the understanding but thought thinking itself (i.e. the Absolute).
I realize the Hegel part is pretty unclear by itself, but the basic idea is that for Hegel, the list of categories is a stepping stone. If we look carefully, we will find problems with each of them (for instance Kant's followers came up with different lists), and we'll find cases they don't manage. As this process continues, we will get ever changing lists until we finally hit upon the perfect way to parse our reality, and that will for Hegel be the Absolute and the Concept. (the Absolute being what we will think about and the Concept being how we will think about it; though we will be thinking about ourselves too because we're part of the Absolute).
I agree with those who say the idea/concept distinction is not one that can be pinned down outside of a particular discourse (or even book).
To give one example usage, Auxier and Herstein discuss the two terms in their book on Whitehead, The Quantum of Explanation. They point out that Whitehead differentiates between the order of ideas and the logic of concepts. Whitehead titles one of his books Adventures of Ideas. In it he lists seven large ideas that have ordered and shaped history. It could not have been called Adventures of Concepts. The distinction here places ideas at a broader level, and assigns them a historical character - think of the 'ideals' of a time - whereas concepts are finer grained and are what govern an individual's particular experience. They write for example that:
[Ideas] are inexhaustible and serve as the ground for an indefinite number of “concepts” which determine our experience (including the “concept” of nature)— for cognition, for action, and via feeling, for reflection. (221).
If you read Shaviro's book Without Criteria, on Kant and Whitehead, he is more apt to draw on Kant's distinction between ideas and concepts, by quoting Kant's assertion e.g. that "Ideas, in the broadest sense, are presentations referred to an object . . . but are such that they can still never become cognition of an object.” Aesthetic ideas are “inner intuitions to which no concept can be completely adequate.” Ideas here are intuitions, whereas concepts are the products of the faculty of understanding.
In both these cases, loosely speaking, ideas are regulative and operate at a broader level of generality than concepts, which are operative and do the actual work of cognizing something (i.e. concepts arise as ideas are put into determinate effect). But by no means are the two terms "settled and accepted" - its a matter of close reading on a case-by-case basis, keeping an eye on the inevitable debates that arise in the secondary literature... What precisely does Kant mean by an "aesthetic idea" or Whitehead by a "logic of concepts" etc?
I have long pondered this question specifically in reference to mathematics. What makes the most sense to me is to consider that an "idea" refers to an approach to accomplishing something. For example,once it occurred to someone that such notions as "half" and "third" etc. are actually quantities like the counting quantities and not just proportions, there arose the question of how to denote them. We can think of a number of possibilities to express some number of equally sized subdivisions; but the one we use is the ingenious idea of a "formal division," i.e., a division that does not indicate an operation. This is a very ingenious idea. Eventually it was realized that it actually is a kind of actual division, that after all there ARE a number of 2's in 1, mainly 1/2.
The word "concept" seems best to refer to an image in the mind which could include a belief that has no basis in reality (which is true of many concepts).
The answer to your second question is easy and complex, but also more significant than the first: "No."
This is because "settled and accepted" (whatever that may mean) in philosophy seems an impossibility on the basis of its necessary disciplinary assumptions/constraints. It is similar to asking a room full of philosophers what "philosophy" means. Can there be a settled and accepted definition of philosophy? And if there is in that room, because we took a poll of these philosophers and they reached consensus regarding the definition, does that mean it can then be generalized to all philosophy/philosophers?
I'm not trying to trivialize here. I sincerely care about this issue, and try to do my best to respect it in each case. To define the philosophical distinctions of any set of terms requires locating them in a particular discourse with a relative placement. I think this may even be one of the most crucial disciplinary responsibilities of the philosopher, though others may disagree.
Therefore, to "respond" rather than "answer" your first question: the distinction between "idea" and "concept" will be contingent upon the particular discourse in which the distinction arises.
A concept is knowing how an entity works and operates for functionalities. An idea is thinking of something to do