Independent clause markers

Here's a short list (as you can see, a complete list would be impossible) of these adverbs, or conjunctions, or <insert POS here>s. It isn't really important how they're named; nomenclature varies from person to person like handwriting or vowel length. And there is no definitive list of Parts of Speech, anyway. What's important is how these thingies work.

also, anyway, as we were saying, consequently, finally, first, fortunately, furthermore, hence, hopefully, however, ideally, in spite of this, incidentally, indeed, instead, likewise, meanwhile, moreover, nevertheless, nonetheless, next, now, on the other hand, otherwise, regrettably, second, similarly, so, still, surprisingly, thankfully, then, therefore, third, thus, unfortunately, well, wherever we go,

Essentially, these are formulaic linking phrases (or clauses, or clauses reduced to phrases), which indicate how two clauses fit together in a context. Mostly those are, as you point out, independent clauses. However, independent clauses normally aren't marked especially. Ordinary sentences, for instance, are all independent clauses, but they have no "independent marker"; it's taken for granted, or -- as linguists call it -- unmarked in the syntax.

Normally it's the dependent clauses that need to be marked, lest they be confused with the unmarked independent ones. And also so that one has at least a prayer of discerning in which of the myriad possible ways the dependent clause "depends on" some other clause. There are a lot of kinds of dependent clauses, like complements and relative clauses, which have quite a lot of marking; and adverbial clauses, which normally use these thingies, too, though with somewhat different syntax.

Some examples of thingies that mark a clause as dependent, and also indicate the manner of its dependency, include:

after although how if once since lest because before than though till unless until when whenever where wherever whether while why

Notice no commas between. Generally, these are integrated into the clause syntax, rather than sitting out if front with a comma parenthesis like one finds with the linkers above. I've put all the single words in the list just above. There are a lot more thingies, but they're phrases. Some examples (separated by | because they usually don't use commas) include:

as if | as though | as long as | as much as | as soon as | as far as | inasmuch as | insofar as | even if | even though | no matter how | in that | in case | in order [that] | now that | so [that] | provided [that] | supposing [that] | given [that]

Notice how many as's and that's there are, and how many wh-words and prepositions are involved. These are idioms composed of spare parts and they're inserted whenever one needs a patch of a certain kind of meaning.

This doesn't really explain much about the various ways these thingies work, but at least there's a list or three of them. That's always a good start.

The word consequently is an example of what the Chicago Manual of Style terms a final conjunction:

Final or illative coordinating conjunctions denote inferences or consequences. The second element gives a reason for the first element’s statement, or it shows what has been or ought to be done in view of the first element’s content. The conjunctions include consequently, for, hence, so, thus, therefore, as a consequence, as a result, so that, and so then [he had betrayed the king; therefore he was banished] [it’s time to leave, so let’s go].

The function of a final conjunction is independent of its position in a sentence.

To understand what’s going on, let’s forget the punctuation and word order. Take the two sentences, The bus was late. Many people consequently missed their appointments. In the second sentence, consequently looks like an adverb of manner, an adjunct, modifying missed. But is it? Adjuncts take their meaning from within the clause in which they are found, as would be the case if we substituted subsequently for consequently: Many people subsequently missed their appointments. There, the adverb clearly modifies the verb and nothing else. It means they missed their appointments at some later time.

Consequently is different. It doesn’t tell us anything about the manner in which or the time at which people missed their appointments, but relates the second sentence to the first. Missing appointments is a consequence of the lateness of the bus and consequently is a type of adverb known as a conjunct. In the words of ‘The Cambridge Guide to English Usage’, 'conjuncts . . . play a cohesive role between separate sentences, or clauses. They . . . express logical relationships such as addition, contrast and causation.’ That seems to me an altogether more helpful description than 'marker'. In both of the OP’s examples consequently plays the same role. As a matter of style, I prefer the punctuation in (b).