I'm not sure what the poster of this question Ezio is looking for exactly. He mentioned that he wants "an updated version that is shorter and has actual calculus but still retaining the purely geometric style" and that Newton's Principia is so elegant because it uses only geometry without coordinates.
To be accurate, Isaac Newton used a form of elementary coordinate geometry in the Principia. The words "ordinate(s)" and "abscissa(s)" are mentioned many times in the Principia in relation or in reference to corresponding geometrical figures.
Let's consider the following figure from Section 2 of Book 2 of the Principia (taken from the English translation by Andew Motte, and the first American edition, which can be found at the Internet Archive site):
Newton mentions abscissas CB,CD,CE, and ordinates BG,CH, and DI.
Here is another figure from Section 5 of Book 1, where Newton mentions the abscissa AD and the ordinate DG:
In other places, Newton mentions the word "axis" or "axes", sometimes in relation to lines representing the (Cartesian) coordinate axes of ellipses or conic sections.
So it can be seen that even Newton's Principia with its "purely geometric style" uses some form of coordinate geometry. I think it is not realistic to find an instructive book similar to the Principia with calculus and without any mention of coordinates or coordinate geometry.
In light of the considerations and remarks above, a helpful book would be Newton's Principia for the Common Reader by Chandrasekhar, as mentioned in the answer by AnOrAn.
Another helpful book published in 1855, using analytical methods and calculus with a style somewhat similar to Newton's Principia, is Analytical view of Sir Isaac Newton's Principia, by Henry Lord Brougham and E.J. Routh, which can be found online at this link.
Hope that helps.