While seemingly not making too much sense as TO GET ACQUAINTED or as INTIMATE,「motions of a hatchet」swung as within the reach of the blade may actually describe a fairly common use of a hand axe, particularly during the bygone era of fewer laws and regulations and prior to the invention of the handgun: in modern times, in the United States of America as an example, statistics indicate that 80% of murders are committed by a relative or a friend (that's 4 out of 5…「intimate」and「acquainted」respectively) 　[　https://bjs.ojp.gov/content/pub/pdf/mf.pdf 2023-05-18　]　
　Frankly, if one can physically sense「pawing cat-like motions, prick teasing」then that INTIMATE sort of activity while GETTING ACQUAINTED must be occurring NEARBY, IN CLOSE PROXIMITY, assuming for the moment that the person sensing these activities by sight, sound, smell, or feeling physical motions is also not personally engaged in them and hence also so distracted.
　For comparison: 'motion of a sword' (辺, or「reaming a foot」) means 'vicinity,' with the sword allegorically being that of the longer reach of the sword of authority of the feudal lord (or, the service provider's legs)…swords are generally longer than hatchets, hence the area covered is wider. Prick-teasing also requires a closer presence NEARBY, with the service provider's face INTIMATE and IN CLOSE PROXIMITY to the prick much more than merely masturbating with the outer extremity of a no-longer-integrated foot.
　As depicted in erotic Chinese art, men in dynastic China masturbated by wrapping no-longer-integrated feet around an erect penis ostensibly to lower the risk of contracting a sexually transmissible disease or infection that often occurred when performing coitus within some sort of mucous membrane system of a female service provider, and, to gain hand-control over the pressure applied to the foot meat surrounding a weakened penis that functioned as a living cockring.
　「Bound feet were once considered intensely erotic in Chinese culture. …The process was started before the arch of the foot had a chance to develop fully, usually between the ages of 2 and 5. …Qing Dynasty sex manuals listed 48 different ways of playing with women's [no-longer-integrated] bound feet」(after removing the binding cloths, of course)
　　[　https://en.wikipedia.org/wiki/Foot_binding 2012-12-17　]　
　Perhaps depicts a household serf cutting firewood NEARBY…but「axe motions」may or may not be near in such cases.「Motions of cat-like hatcheting, subtly cruel and malicious」describes human feelings in many cases that extend to one's neighbors NEARBY, the stuff that high fencing, festering feuds, and long wars are made of.

Modern definitions (that generally disregard history) …excluding politically incorrect concepts and other meanings deemed offensive today; may list only pigeonholed definitions, euphemisms, or meaningless mnemonics)
Chinese: near; close to; approximately
Japanese: near, close, short, neighbourhood, neighborhood, vicinity, nearly, close to, shortly, soon, nearness, before long, lately, recently, nowadays, intimate, acquaintance, to approach, to draw near, to get close, to get acquainted with, to get closer to, to get to know, to bring near, to put close, to let come near, to associate with
Unihan extended: near, close; approach; intimate
EDRDG: near; early; akin; tantamount

Used within names (nanori)
おう　･　おお　･　この