The axles on which a wheel spins are derived from the "axles" on which the spheres in the heavens spin and around which they rotate. Thus an axis is a specific use of this axle notion, for it is noticeable that the gyroscopic properties of spinning bodies determine a fixity in space, which though dynamic is always fixed in orientation. This fixedness is therefore less arbitrary than a spiraling motion path and is the preferred "basis" for "measuring from".
Several ,at least 2 rotating axes, arbitrarily relative are needed to fix an orientation frame in space. In addition two arbitrary seemeia if lying on one axle always have a fixed relative orientation, and should at least one seemeion of a pair on an axle coincides with another seemeion of a pair on another axle and maintains that coincidence, then a fixed relation exists between the axles relatively. Alternatively should one seemeion of a pair always move dynamically on another axle then the axles are in dynamic motion relatively.
Finally should the compatriot of such a seemeion move in rotation around its fellow while its fellow moves, or remains fixed along another axle, then a complex rotational motion of the axles is possible.
Of course, there are many other relative motions which do not have the fixity or rigidity of axle motions, but our experience so far is that density has this attribute inherent in its inertial behaviour, and other types of behaviour involve causal interventions we may call "pressure", which may be internally generated or externally generated,
Thus axles from the latin or the greek have the underlying notion of driven or pressed behaviour, fixing the motion by some ordinance, and requiring some other pressure or drive to move out of place.
The axes of a reference frame typically do not convey this dynamic nature, and so once again our formalism has cut us off from the greek dynamic view and experience.
As relates to axioms, the notion of an impelled idea, one thrust upon the student or participant in an argument or debate again is translated tamely as that which is agreed, rather than submitted to. Whereas a lemma is a notion one willingly accepts for profit or benefit of accepting it, as it may happen .