I think the SEP really tries its best to be understandable for newcomers while not dumbing-down things too much (unlike so many popular philosophy books and articles). As a non-philosophy grad trying to intermittently delve into philosophy, the encyclopedia has been a very valuable resource to me, for getting a rough outline of the philosophical ideas and debates throughout history.
"As in most philosophical movements, not all positivists agreed with each other, but they generally agreed that if you couldn’t verify something, it was meaningless."
Here is Ney on positivism (p. 121):
"Logical positivists differ on what verification must involve in any particular case, but a pillar of logical positivism was the view that there are two basic kinds of verification: by analytic and by synthetic methods."
Or Godfrey-Smith (p. 27):
"I turn now to the other main idea in the logical positivist theory of language, the verifiability theory of meaning."
You can also look at Devitt and Sterelny's discussion of verificationism, or the section on logical positivism in Schwartz. They are pretty consistent with the quoted paragraph from Philosophy Bro.
The short answer is that logical positivism was always extremely diverse, and there's no reason to think that there's a consistent core to it, but if there is then it has to do with a belief in making progress in philosophy and science by (a) using logic to build on (supposedly) certain foundations (although they didn't agree on what those foundations were), and (b) repudiating metaphysics (which meant different things to different people, but which IMO was probably the most important point of agreement among the logical positivists anyway).
Verificationism was fairly popular among logical positivists, especially at the beginning, and especially in English-speaking countries, because Ayer, who wrote an early and very popular book about it in English, stressed verificationism. But Ayer's book was over-simplified to start with (IMO), and in any case logical positivism continued to develop, with almost all of its important literature being in German.
All the major logical positivists (I think - certainly most of them) sooner or later - and in some cases very soon - gave up on verificationism, but still called themselves logical positivists (or logical empiricists, which was and is usually thought to be more or less synonymous).
From the 1930s onwards, English became more important than German for the first time, and the most important logical positivists were then those who were in English-speaking countries, especially Carnap in America (and arguably Wittgenstein, depending on whether you think he'd been a logical positivist to start with). All of them (I think, and certainly Carnap) repudiated verificationism pretty early on.
It's all super interesting and I encourage you to read https://plato.stanford.edu/entries/logical-empiricism/ :-)