Please help me to better understand the simile of the turner for the first part of the anapanasati instructions

I suppose the word is related to Bhamati

Bhamati Bhamati [bhram; on etym. see K.Z. iv.443; vi.152. Expld at Dhtp 219 by "anavaṭṭhāne," i. e. unsettledness] to spin (of a wheel), to whirl about, to roam Dh 371 (mā te kāmaguṇe bhamassu cittaŋ); J i.414; iii.206= iv.4 (cakkaŋ matthake); iv.6 (kumbha-kāra-cakkaŋ iva bh.); v.478. — pp. bhanta. — Caus. bhameti to make whirl Vism 142 (cakkaŋ).

A "turner" (in English) is someone using a lathe. There's a photo in the Wikipedia article of someone using a wood-lathe -- there are surely lots of YouTube videos.

While the object is spinning, you introduce a sharp chisel, to make a deep or shallow cut.

Lathes were used in ancient India.

Please see the YouTube video "Easy Wood Turning Lathe Projects" to understand what modern wood turning is about, and what a modern lathe looks like.

I also found an interesting YouTube video entitled "Ancient Lathe Machine Found in Hampi, India - Lost Technology Discovered?" which shows how a lathe may have been used in ancient India.

On the same YouTube channel, there's another video on this same topic - "Hoysaleswara Temple, India - Built with Ancient Machining Technology?"

It's possible that it was used not just to carve wood, but also stones, according to these videos.

The Pali terminology referred most likely to a carpenter's tool, where putting the tool in motion required long or short manual movements. The term añchati can refer to either a linear or a circular motion, so the important aspect is to understand the mental implications of the simile.

An interesting free-flow explanation was given in a commentary by paraphrasing the idea and expressing it in a different scenario for a modern audience as: Just as an experienced traveller, while taking a short route he acknowledges “this is a short route” or while taking a long route he acknowledges “this is a long route”, just so, a monk while breathing in long, he acknowledges “breathing in long”; when breathing out long, he acknowledges “breathing out long”. When breathing in short, he acknowledges “breathing in short”; when breathing out short, he acknowledges “breathing out short”.

What was interesting about the commentary was not only the easier to understand simile but also the explanation that be it a carpenter, a potter, or a traveler, regardless of the metaphor, they all imply decision-making behind the action itself so it can prove useful to control the breath at first, alternating short inhales and exhales with long ones, until the attention is able to focus on the breathing process as it occurs autonomically.

Last but not least, as the attention becomes focused, He trains the mind “with every in-breath there will be the same awareness for all bodily sensations” and he trains the mind “with every out-breath there will be the same awareness for all bodily sensations”. He trains the mind “with every in-breath there will be the calming down of bodily reactions” and he trains the mind “with every out-breath there will be the calming down of bodily reactions”.

The lathe is truly an ancient tool. I do not know how the lathes of ancient India might have been different from this one in this video (besides materials used to make it). This is a bow lathe; another similar lathe is called a spring-pole lathe. (I do not belief flywheel- or “great wheel-“ lathes were invented yet). So more likely than not the piece of wood the turner worked rotated both forward and backward, meaning the turner can only cut half the time.

It’s a very vivid simile: a turner’s attention is concentrated on the edge of their gouge or chisel, and the workpiece at (and immediately near) the place the tool will contact it. It “deserves” close attention, because incorrect application of the tool could quickly ruin the workpiece and the tool might even “catch” the workpiece. Neither a long- or short-cut is inherently better than the other.

Interestingly, as an aside, lathe-turning has been referred to as “The hobby of kings” (At least in the West). It is generally considered an intrinsically gratifying past time.

Hope this is helpful. Similes are often only as useful as they are familiar; this is my best estimate of the kind of turner described in the suttas.