Species aren't necessarily as well defined as you might like. The (complete form of the) usual definition is something like, "A species is a group of critters (that look alike, and) that interbreed among themselves but which do not (normally) interbreed (in the wild) with members of another group of critters (and produce fertile offspring)." (Plus a few caveats that I've probably missed.) There are many reasons why two purportive species do not interbreed: they physically can't, they produce sterile offspring, they don't find each other attractive, or they are simply geographically separated; although in the last case, separated groups tend to quickly become morphologically different, in which case one of the other options applies.
For a brilliant example of the taxonomic follies, check out ring species[1]:
"The Lesser Black-backed Gulls [of Northwestern Europe] and [European] Herring Gulls [mostly from Great Britain and Ireland] are sufficiently different that they do not normally hybridize [, but the E.H.G. can hybridize with the American Herring Gull, (living in North America), which can also hybridize with the Vega or East Siberian Herring Gull, the western subspecies of which, Birula's Gull, can hybridize with Heuglin's gull, which in turn can hybridize with the Siberian Lesser Black-backed Gull]; thus the group of gulls forms a continuum except where the two lineages meet in Europe."
This is not to say that "species" is not a meaningful distinction, is a "judgement call", or is "open to personal opinion" or something. There are good reasons to divide critters up, but unfortunately, it is not the case that Someone sat down and said, "THESE THINGS ARE NORMAN'S HERRING GULLS, NOW AND FOREVERMORE!"
"We've observed finches turning into finches (with different beaks)..."
Finches are a family (as in kingdom-phylum-class-order-family-genus-species) containing something more than 100 species.[2] Please don't make the creationist's error of declaring, "all bats are bats and are all one thing". If a different beak shape makes Rupert's Seed-Eating Finch unattractive to the Red-Breasted Yellow-bellied Fruit-Smacking Finch, so that they don't typically hook up, then they're legitimately different species, as much as any two others.
There is no other explanation besides natural selection that could explain these basic facts. Besides Darwin other scientists were independently converging on the exact same solution.
In the same way only natural selection could explain the existence of vestigial features among related families of species such as the existence of hind limbs in whales or the curious why that the eye of the vertebrates is constructed. This was recognized immediately in the 1860s once Darwin's work was published.
DNA evidence supports evolution 100%. Hypothesized descent from two families can be easily verified from DNA evidence. The whale really once had an ancestor that walked on the land with four limbs. There is simply no other explanation could produce DNA in the exact way that would be predicted from natural selection.
This is just the tip of the iceberg.
Take the present fossil ostensibly under discussion here. Was that in fact an intermediary creature between species? I don't know, maybe. I do not see compelling evidence that we know for sure. I for one really have a hard time stating such things as facts with a straight face; I feel like I need to prepend phrases like "The fossil record suggests that".
Incidentally, I feel the same way about historical anthropology... statements like (making something up) "These ancient people wore fur hats and ate with their fingers out of stone bowls," based on some bowl fragment and a bit of fur found near the remains of a skull. Well, I don't know, maybe that was true! But such evidence seems weak to me, and hardly enough to present those ideas as facts.
And perhaps my gripe is more with mainstream media presentations and with loosely-written educational materials than with actual scientific research.
Maybe the problem is that you don't have the full context of the evidence being presented. How was the material found; was it in the exact same layer of dirt? Was it dated to the time period of the rest of the site? Some bowl fragments don't provide evidence independent of the context that gives them validity.
But you should be comforted to know that the evidence for evolution from DNA is so overwhelming that there is no room for doubt at all. Species are related to each other and the approximate time that their common ancestor lived can be established as well independent of any fossil evidence. There is no other plausible hypothesis besides evolution that can be presented to explain the evidence.