I regret I do not have a copy of Kiselev at hand for this commentary, having given mine away after becoming dislliusioned with it, but from searching online I offer another possible example that is especially important to me. It seems, that Kiselev, like many other people, defines a line as tangent to a circle if it meets that circle in only one point. (This information comes from a homework set on Kiselev and not from the actual book, but if correct, it is to me a significant flaw in Kiselev compared to Euclid.) This definition is not Euclid's and has almost no value in terms of understanding the concept of tangency. It applies only to quadratic curves, as opposed to Euclid's which applies to all curves at all except inflection points. Namely Euclid deines a line as tangent to a curve if it meets it without "cutting" (across) it. This is indeed the basic idea via intersection theory of a line meeting a curve with multiplicity two (or any even multiplicity).
Then Euclid proves the more usual result that a tangent line to a circle is perpendicular to the relevant radius, and more interesting, if given any other line through the same point, the circle eventually interposes itself between that line and the tangent line. If you think about it, this is the limit definition of a tangent line in terms of secant lines. From what I have read, Newton may have read Euclid just before giving his own version of this definition, and in any case Euclid's definition and theorem prepares the mind for the general modern definition. If you have Kiselev at hand, perhaps you can give us more data on this concept as treated there, in particular whether he includes this fundamental limiting version of tangency.
Another interesting result in Euclid, which is not always credited with containing any trigonometry, is the geometric version of the law of cosines, Props. II. 12 - 13. These are clearly presented as simple generalizations of the Pythagorean theorem, but most of my students have had no familiarity with them, or often do not realize that the analytic versions they have seen are just that.
I should admit that in teaching it, I also alter Euclid a little in Book II, introducing a bit of algebraic notation for the (to me) somewhat tedious geometric statements that things like A(B+C) = AB + AC, and I make explicit the idea of a geometric multiplication, in which the product of two segments is a rectangle.
But the fun of geometry is that there are many ways to try to present it, all of them interesting in their own way. After trying many of them over a lifetime of teaching I just prefer Euclid's now that I see how it is foundational to so many modern ideas. In Book V e.g. we see clearly for the first time in history the idea, now often credited to Dedekind, of real numbers (arbitrary ratios of segments) as things approximable arbitrarily closely by rationals.
FWIW my notes on reading thr first part of Euclid are here:
http://alpha.math.uga.edu/~roy/camp2011/10.pdf