Borrowing from another post: Are Newton's laws just definitions?
"Newton follows the Euclidean tradition of presenting mathematical proofs by first providing a set of definitions which are then followed by a set of axioms (i.e., laws) that are assumed to be true. Definitions in themselves do not admit the existence of anything, but rather provide terminology that is later recognized as a particular condition as a result of a set of axioms.
Newton presents his laws of motion based on a collection of known observations regarding motion (in particular, he cites Galileo's inclined plane experiment and hypothesis that objects only slow down due to air resistance or other friction forces, which is counter to Aristotelian physics), and carries forward with the logical consequences of these laws in a series of propositions, which so happens to match experimental evidence.
Once the laws are provided, the particular effects of the laws can be matched with the definitions he provided. For example, his definitions of quantity of matter, quantity of motion, vis insita, etc., only make sense within the framework provided by his laws.
...the laws are not "definitions" since definitions are only understood within the assumed truth of the laws."
In other words (and to directly answer your question), "Definitions" provide a specific term used to compactly identify something that arises from some set of axioms or propositions.
As a loose example, in calculus, we define an "inflection point" as being a specific point where the second derivative of a function equals zero. This definition does not declare the existence of the inflection point as an axiom or proposition but instead provides an identifying term for when it does occur based upon a set of assumed truths.
In Axiom 1 Newton defines his concept of inertia but he classifies his
definition of inertia as an axiom (not as a definition).
Newton's First Law (i.e., Axiom 1) does NOT define inertia, rather, it declares an axiom of motion that is assumed to be true. The term "inertia" is nowhere mentioned in the law: "Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impress'd thereon." Wikisource: Newton's Laws.
However, in the "definitions" section that's located prior to his Axioms, Newton defines inertia (Vis inertiae) under Definition 3: "The Vis Insita, or Innate Force of Matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forwards in a right line. This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the Mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this Vis insita may, by a most significant name, be called Vis inertiæ or Force of Inactivity." Wikisource: Newton's Definitions
Thus, "inertia" is a term used to identify the effect (namely the Force of Inactivity) that results from axiom 1. However, "inertia" does not rely on axiom 1 alone because the definition states it "is a power of resisting," thereby invoking an axiom of change (i.e., Law #2) and that it is categorized as an "Innate Force of Matter," which practically speaking, can only be understood through some interaction (i.e., Law #3 where inertia is perceived as a reactionary force in response to a force). Naturally, you might ask, "then why is Law #1 called the 'law of inertia'?" Well, because inertia is still "there" during the absence of any change and external forces; however, if we omit Laws #2 and #3 and place Law #1 in a vacuum, then inertia becomes an absurd concept since there are no rules that allow change and forces that permit change. In other words, the definition of inertia is absurd unless we admit all three laws.
And also, the title of the chapter is, "Axioms, or the laws of
motion." So Newton, defines his axioms as laws of nature.
Any thoughts about the difference in meaning between definition, axiom
and law in Newton's time?
Just as you said, Newton titles the section "Axioms, or the laws of motion (Axiomata sive leges motus)." Thus, Newton treats "Laws" as being synonymous with "axioms." If I were to guess, he probably chose the term "Laws" in order to categorically distinguish these rules of physical governance from mathematical axioms, but they are functionally the same. It's much like "postulates" versus "axioms" in maths in which they are functionally the same, but "postulates" is a species of "axioms" that act within a specific field of study.
For more examples of how definitions are treated, look at Euclid's Elements Book 1: Definitions are provided first, followed by postulates (i.e., axioms), then common notions, then propositions. Though definitions are provided first, they are only fully understood within the context of the postulates and constructed propositions. Elements, Book I