'Analytic philosophy' has evolved into an equivocal term. At one point in time it denoted a somewhat cohesive movement in philosophy, including the likes of Frege, Russell, Wittgenstein, etc. It involved a self-conscious rejection of the Hegelian influence in Britain and America and it rejected speculation, metaphysics, etc., instead striving to ground philosophy on logic and regulate philosophical statements within a formalized language devoid of the kinds of ambiguity that Russell discovered when he read Hegel in the original.
What cannot be ignored is that parallel with the development of early analytic philosophy there have also been strong countercurrents that existed in America and Britain. Contrary to what a generation raised on Russell's fraudulent History of Philosophy believe, Hegel never really went away in the US, nor did Kant. For example, there is the tradition of American pragmatism, developed by the likes of Pierce, William James, Dewey, Mead, and so on. All along you have Sellars who is secretly graduating analytic philosophy to its Kantian phase, and Ryle who smuggles Husserlian phenomenology into Britain without declaring it at customs. Slowly these subversions have gone mainstream: In my view, Quine, Kripke, and Rawls represent three decisive changes in the dominant image of analytic philosophy, especially when you pit them against David Lewis' Humeanism. Quine is where the collision between pragmatism and analytic philosophy occurred. Kripke opened analytic philosophy back up to metaphysics in a new style (Kripke, by the way, had secret sources for his modal logic coming from Stig Kanger in Europe, who by turn was influenced by Polish logicians, who in their turn, were coming off of Husserl, Twardowski, Meinong, and others). Rawls introduced what is now analytic practical philosophy. From then on, the definition of what can be considered analytic has been expanding, and has still not realized its full effect. Recent analytic philosophy has absorbed many of the countercurrents that the analytical movement was founded on rejecting, and what is today called analytic philosophy barely looks anything like the original movement, with philosophers such as the Hegelian pragmatist Robert Brandom, the Kantian Korsgaard, Chalmers with his bizarre (speculative!) mix of Frege and Carnap, Aristotelians abound, pansychists and Spinozans on the rise, Buddhist philosophers, and even transcendental phenomenologists like Zahavi sneaking into the fray through the backdoors of philosophy of mind, along with a host of radical neo-Kantian neuroscientists.
Almost none of the so-called 'analytic' philosophers these days can be recognized as being involved in the early movement mentioned prior; virtually none have faith in traditional logicism or logical atomism or positivism; logic-oriented philosophers worship Frege but accept none of his doctrines; relatively few are strict followers of Quine, Kripke, or Stalnaker, and the so-called "linguistic turn" resolved into philosophy of mind. Curiously, late-period Wittgenstein is the exception that proves the rule. There are a lot of straight-up Wittgensteinians out there, especially in epistemology where they have migrated after being largely ousted from logic and mind (though there are pockets of Wittgensteinian logicians who still pursue the Tractatus). Analytic philosophy is more defined nowadays, if by anything, by its strict division of labor, increasing specialization, and lack of references to Deleuze and Derrida. But even these currents are being resisted by followers of Evans and Brandom, etc., who are more systematic. It's nearly impossible for any one person to get their head around what's haphazardly grouped under analytic philosophy today, and my sense is that history will see this as a period of transition where historians of philosophy will no longer see much use labeling what's going on in the early 21st century with the same term as what a handful of philosophers who were reacting against Hegel in the early 20th century were doing.
All of this goes towards answering your questions.
Bertrand Russell's debate with Fredrick Copleston and they mentioned Leibniz's division of propositions. I looked into it, and it seems, based on limited research, that no one seems to take pre-1960s analytic philosophy very seriously.
I think you're right about that. As I detailed above, early analytic philosophy is quite dead. Many important concepts are named after them--e.g., "Russellian" propositions, "Russell's" theory of direct description, etc. But rather than signalling that we are taking Russell's philosophy as a whole seriously, this is just shorthand. The early analytic philosophers are considered immensely influential and are given due credit for their innovations, but by and large this is a period in the history of philosophy now.
I'm unaware of what modern analytic philosophy would look like, and I'm interested to read the main players.
Who are the serious analytic philosophers; or, at least, the philosophers which are taken seriously?
As I detailed above, it's a messy terrain principally because it's happening now and we can't get a history of it. If you want to get the latest crop of what are undeniably analytic philosophers you'll want to look into Quine and Kripke and their direct descendants, as I mentioned above. You have to think in terms of lineages instead of the monolithic 'analytic philosophy'. There's a database called the Mathematics Genealogy Project, where you can enter a mathematician's name and see their "family tree" of teachers and students. This is how academic disciplines work, one gains influence by the dissemination of one's influential students. I wish there was one for philosophy. Some philosophers are in the mathematics database, depending on if they have any connections to mathematics and logic. As for the main players of today's analytic philosophy, the question is---main players of which subfields? There are a lot of big names out there, some of which I dropped above, but they're all doing very different things. Keep your eye on lineages to clarify a philosopher's orientation and aim.