It seems like there are mixed results because sometimes I read about a single missing link, like an archaeopteryx that somehow single-handedly explains all modern-day birds, but then I see conflicting articles about how different birds are descended from different dinosaurs like a t-rex or a velociraptor and so on. Which theory is correct? Did birds all descend from one common ancestor or multiple?
The answer is "one common ancestor", but I'll expand.
All organisms descend from one common ancestor so that question is not quite well-posed, but what you are actually asking I think is whether birds all descend from one common ancestor that was a bird, or whether their common ancestor wasn't a bird, which implies that different branches of birds became birds independently. In other words, are birds a "polyphyletic" group or a "monophyletic/paraphyletic" one (the difference between the latter is whether all of that common ancestor's descendants are birds, or whether it also had descendants that aren't birds).
The answer to that is that modern birds are monophyletic: they all descend from a common ancestor that, itself, was a bird. (and that common ancestor doesn't have any descendants that aren't birds)
But modern birds aren't the whole story - their group originates in the Cretaceous, and there are many groups of birds that are clearly recognizable as birds but also aren't modern birds - they might have teeth, they have subtle but unavoidable skeletal differences, etc. Like Enantiornithes and Confuciusornis. In other words, modern birds absolutely, unambiguously have a common ancestor that was itself a bird, which isn't hard because animals we would recognize as "birds" were already common by the time it appeared and the ancestor of our modern birds was just one of them. Whether those older, "extended" groups of birds themselves descend from a single species is harder to tell because fossil evidence is all we're going on, and it's even harder to tell when we go closer to the origin of birds and they no longer look unambiguously like birds. But at that point it's not so much a matter of "did this group originate from one ancestor or several" but "are these fossils that we've been putting in this group as closely related as we were assuming?" and "did this trait that these different fossils have evolve in their common ancestor, or did they evolve it independently, and what about those other fossils that seem related but don't have the trait: did they lose it from the common ancestor who had it or are they a sign that the common ancestor didn't have it to begin with?". Like, you might get different answers depending on how you define "bird". BUT paleontologists these days like monophyletic groups, so whichever groups they call "bird", those will have a common ancestor that was a "bird".
The Wikipedia page for Avialae expresses some of the issues with what "bird", or "Aves" means in the context of paleontology. To quote:
- Aves can mean those advanced archosaurs with feathers (alternately Avifilopluma)
- Aves can mean those that fly (alternately Avialae)
- Aves can mean all reptiles closer to birds than to crocodiles (alternately Avemetatarsalia [=Panaves])
- Aves can mean the last common ancestor of all the currently living birds and all of its descendants (a "crown group"). (alternately Neornithes)
"Avialae" is one of those "extended groups of birds", in that paleontologists will often refer to any member of that group as "birds", but there's still a wide gap between "Avialae" and "Neornithes"; some stepping stones on that gap:
Neornithes (modern birds) are part of
Euornithes (birds have a certain articulation oriented like modern birds, as opposed to Enantiornithes, or "opposite-birds", who have it the other way, in addition to other differences like having teeth; note this page has a cladogram), which is one branch of:
Ornithothoraces (all descendants of the common ancestor of Euornithes and Enantiornithes), which is part of
Avebrevicauda ("short-tailed birds", to distinguish them from long-tailed Avialans like Archeopteryx) which is part of
Avialae, which as mentioned is basically birdinosaurs that could fly like Archeopteryx but it doesn't stop there because that's part of
Paraves, a group of dinosaurs that by and large had wings and feathers including the four-winged dinosaurs like Microraptor gui, which if we saw today we'd recognize as not like other birds but would we recognize as not a bird, really? That's part of
Pennaraptora, which is the first group to contain names that we definitely think of as not-birds; this group contains birds as well as Oviraptor and Deinonychus, but to quote the page:
The earliest known definitive member of this clade is Anchiornis, from the late Jurassic period of China, about 160 million years ago.
And Anchiornis, as its name suggests, is bird-like, feathered wings and all. This is where we get into "wait, Velociraptors were basically like turkeys?" reactions (Deinonychus is Jurassic Park's "Velociraptors").
(also these points are where the groupings become disputed, or fast-changing, or ambiguous between groups that are defined based on phylogeny, or similarity, or others and you can get different results depending on which group's Wikipedia page you look at; for example the Wikipedia page for Theropods puts Avialae directly under Maniraptora).
Pennaraptora is part of Maniraptora, which also contains your velociraptors, and Maniraptora is part of Coelurosauria which also contains your Tyrannosaurs. At this point we have clear non-bird dinosaurs, although it's likely they all had feathers.
I don't know which articles you read saying some birds would have descended from T-Rex and others from Velociraptor; that seems completely counter to what we know today of bird evolution, even if we extend "birds" to "Avialae" or even "Paraves". In fact the only way it would work is if we call T-rex or velociraptor themselves "birds". What you could have is conflicting articles about how close birds are to T-rex or Velociraptor; that part of the family tree is based entirely on fossils and so new fossils can change our understanding of how things are related... But the current phylogeny that makes, say, Avialae a group that's quite a few nodes away from velociraptors, and both of those a couple of nodes away from T-rex, seems quite robust. We have found many, many bird and proto-bird fossils in the last few decades that have clarified the picture on that scale (you can get an idea of how many by clicking the various links).
One thing those many, many bird and proto-bird fossils also made clear is that the traits of modern birds (feathers, wings, toothless beaks, etc) didn't evolve in a simple line from non-bird to bird. Many of those traits evolved convergently in several lineages, were lost in some, maybe regained in others, and feathers in particular turn out to be a widespread dinosaur feature that cannot be considered a uniquely bird trait anymore (unless we want to call T-rexes "birds"). Still, saying "beaks evolved several times" or "feathers evolved several times" doesn't mean that birds, let alone modern birds, evolved from several different ancestors. It can mean that the common ancestor of birds had lots of variously bird-like more-or-less distant cousins living around the same time.