While the concept of explicit solution can be interpreted messily, as in the quote above, there is a version of this idea that more closely cuts reality at the joints, computability. A real number is computable iff there is a Turing machine that outputs the number to any desired accuracy. This covers fractions, roots, implicit solutions, integrals, and, if you believe the Church-Turing thesis, anything else we will be able to come up with. https://en.wikipedia.org/wiki/Computable_number
Computability does not express the same thing we mean with “explicit”. The vague term “explicit” crystallizes an important concept, which is dependent on social and historical context that I tried to elucidate. It is useful to give a name to this concept, but you cannot really prove theorems about it (there should be no technical definition of “explicit”).
That being said, computability is of course important, but slightly too counter-intuitive in practice. Say, you have two polynomial vectorfields. Are solutions (to the differential equation) computable? Sure. Can you say whether the two solutions, at time t=1 and starting in the origin, coincide? I think not. Equality of computable reals is not decidable after all (literally the halting problem).
re: differential equation solutions, you can compute if they are within epsilon of each other for any epsilon, which I feel is “morally the same” as knowing if they are equal.
It’s true that the concepts are not identical. I feel computability is like the “limit” of the “explicit” concept, as a community of mathematicians comes to accept more and more ways of formally specifying a number. The correspondence is still not perfect, as different families of explicit formulae will have structure(e.g. algebraic structure) that general Turing machines will not.
Polynomial-time computability is probably closer to the notion of explicitness (though still not quite the same, as daozaich points out). I don’t know of any number that is considered explicit but is not polynomial-time computable.