What is... A Parsimonious History?
Note: [2016-06-01] Please note that OP has changed the title which was:
How can a modern historian interpret historical mathematicians?
and he has added question 2. This answer addresses the original question which is at the time question 1.
To me it seems Jeremy Gray, the author of the referenced book, put some words of warning that interpreting the work of the old masters in terms of one or two modern concepts only is too often superficial and not appropriate to fully grasp what was really going on.
If we look at the last sentence of OPs cited paragraph we can see that the historical development is more than that.
From section 1.5: But it is to say that the foundations of the calculus were for at least two centuries the subject of shifting, partial, and largely coherent speculations that form the opening chapters of the history of analysis.
To better see the author's position I like to cite from Hidden Harmony-Geometric Fantasies: The Rise of Complex Function Theory written by Jeremy Gray and Umberto Bottazzini.
From the Introductory Section: In any history of ideas, the historian seeks to show how things once thought about in one way became thought about in another. As complex function theory developed many ideas were first introduced naively and only slowly refined.
Definitions were lacking, and when provided were sometimes inadquate by later standards. Moreover precision, when it became available, could be misleading: mathematicians on occasion offer a clear definition with very few ideas about its deepest implications - as the example of continuity in real analysis shows.
Sometimes these problems can be confronted directly, as with the very definition of an analytic function, but more often one has to ride out a long period of some vagueness.
Let us note some specific issues: Cauchy, for example, often used the phrases continuous and finite and continuous very loosely to mean something like complex analytic. Similar problems occcur with counting roots according to their multiplicities, with $\lim$ versus $\limsup$, and points of infinity and poles. ...
Later on the authors continue with their preferred approach to the history of mathematics and mathematicians:
There is therefore no truly satisfactory way to represent the originial ideas of mathematicians when they are like this. To say nothing is to produce confusion. To silently bring them into line with modern standards not only introduces anachronisms but also brings in historical falsehoods and nullifies the purpose of history.
To correct them in more than the most egregious cases is to encumbeer genuine blunders and thereby diminish the work of major mathematicians.
The best policy is to read on in a spirit of dialogue with the earlier authors, aware, as one might be, of the limitations and false implications of their papers and books, and waiting to see when, if at all in the period, better light was shone on the subject. In this way one can grapple with more of the complexity, and the drama, of the past.
Comment
What about Detlef Laugwitz's comments in his: Infinitely small quantities in Cauchy's textbooks, Hist.Math. 14(1987), no.3, 258–274 [that you cited elsewhere in SE]:
As a historian of mathematics one cannot but take an author’s own intentions and reasons seriously: Infinitely small quantities are fundamental in Cauchy’s analysis, they are compatible with rigor, and they produce simplicity.
[Cauchy's theorems on continuity and convergence] are incorrect when interpreted in the by now common conceptual framework of analysis (which obviously cannot have been Cauchy’s framework). Both theorems become correct as soon as one adds assumptions on uniformity (which, at least in the form by now common, were never used by Cauchy). The theorems are correct in any of the modern theories of infinitesimals (which, apart from being unknown to Cauchy, lack the “simplicity of infinitesimals,” at least in the version of Robinson).
The three attitudes mentioned (Cauchy erred; Cauchy forgot about essential assumptions; Cauchy was correct, but only when put against a modern background) are unsatisfactory from the point of view of a historian. [...] The only satisfactory attitude should be: Try and understand Cauchy’s theorems and their proofs from his own concepts.
Note: according to my understanding of J.Gray's point of view, a "parsimonious history" is an approach aimed at understanding past theories and concepts "in their original environment", avoiding to overload them with recent developments.