I have a slate floor that has already been sealed, but needs to be
cleaned and polished so there is a low-sheen that accents the color of
the tile.  I do NOT want a high-sheen (i.e., waxy) finish.

Two questions: What is the best approach to doing this that (1) uses
products you can get in any supermarket? and  (2) uses a commercial
"made for slate" -type product? Thanks!

Slate is porous and you want to be careful to use a product made
especially for use on porous stone and not a wax which will clog the
pores of the stone and also produce a waxy buildup.

Here are suggestions from several sites on the care of slate floors.


Also please avoid Waxing a natural stone! Stone is porous and needs to
Breathe! Trapping moisture under wax can actually cause a stone floor
to decompose !
http://www.nvo.com/integrityclean/scrapbook1/

Johnson Wax
A GUIDE TO THE CARE OF MARBLE, TERRAZZO & STONE FLOORS
Polishing:
The use of emulsion based polishes to protect floors is well known.
Whilst polishing does not affect the intrinsic qualities of the stone,
it in itself offers a barrier to etching or staining materials. Whilst
an emulsion polish is not as hard as a vitrified surface, abrasive
materials scratch the polish and not the stone. In essence, polishes
offer a sacrificial layer, which can be readily repaired or, “at
worst”, stripped and reapplied. Polishes may be applied to a stone
floor no matter its condition; that is, the floor may be in an
untreated state, ground or
vitrified. It is recommended that a floor with a high gloss, through
grinding or vitrification, have one or two coats of a suitable “seal”
– Carefree Undercoat applied to aid adhesion of the polish. Indeed, it
is advisable that this seal is used, no matter the condition of the
floor, to improve adhesion and durability.
Furthermore, polishes afford the opportunity of achieving a required
appearance – a high gloss achieved by using a product such as Carefree
Eternum, whereas a satin sheen is obtained through use of Carefree
Satin.
http://www.jwpeurope.com/WEB/013/scjohnson.nsf/8da05c6904e05d06802567fb0055dbfd/827550a01323940e80256af100382878/$FILE/Stone.pdf

Stone Tips
Q. Can you advise the difference in performance between water based
and solvent based color enhancers? Also, I've read about products that
give a high gloss finish to slate and other porous materials (like
Jamo High gloss Sealer). Can you recommend a product that will provide
some gloss but doesn't need to be stripped for re-application?
Answer:
The difference between the two in performance is the solvent based
sealers slightly out perform the water base, and the gap is getting
very close. The advantages to the water base are it is more
environmentally safe, less odor, and can be shipped without hazardous
material labels. Always test the product first to make sure you like
the result.
http://www.natural-stone.com/st_detail.php?id=3222

Never use waxes or One step acrylics, these products result in
temporary gloss, but over time, can cause serious damage and
discoloration. Avoid the use of cleaning products that have
distinctive colors. These used over a period of time tend to impart
their own character into the marble
http://www.olsoncleaning.com/News1.htm

Wash slate floors with detergent and water. Slate is not damaged by
alkali cleaners, but strong solutions are not needed for cleaning.
Rinse well and wipe dry. A water based, self-polishing wax can be
applied over a clean, dry, sealed floor for extra protection and
shine. Weekly damp mopping with a barely damp mop on a waxed floor
will keep it clean for quite a while before the washing with detergent
is needed.
If you have an electric polisher, solvent-base polishing wax may be
used to clean and wax. This must be buffed. Do not use wax around
fireplaces.
http://www.growinglifestyle.com/article/s0/a14240.html



To seal slate; 
We recommend the same basic procedure as listed above, 511
Impregnator, followed by
Mira Matte
. To achieve a shiny slate look, follow the 511 Impregnator with 
Mira Shine
http://www.linley.com/newsspr9.pdf

Slate is most often used as flooring because it's so economical- $14
to $35 per square foot- and durable. Because it's a soft material,
however, it's liable to scratch. Fabricators suggest applying linseed
oil once a year to remove scrapes.
http://www.buddyrhodes.com/news_template.asp?include=News&subsec=chrnstn3

Wincilate Product Data Sheet 4 
Purpose Made Slate Flooring
 Once floor has been sealed it will, of course, require additional
periodic treatment according to the amount of traffic it receives.
Once sealed, the appearance can be maintained by the application from
time to time of a proprietary polish. A polish which has been found
satisfactory over the years is a metalised emulsion.
Once the finish has been brought to the required level, occasional
buffing with an electric floor polisher should maintain the
appearance. Application according to maker's instructions.
It should be noted that all polishes give an increasing shine.
http://www.contour-computers.co.uk/wincilate/flooring.htm

Sites offering sealers and glazes:
http://www.stonesealer.net/prod04.htm
http://www.custombuildingproducts.com/care/Default.htm
http://www.borinquenmarble.com/maintenance.htm
http://www.marbleservices.com/Prod/Prod-Res/Prod-miracle.htm



Search strategy: slate polish; slate floor polish; sealed slate
polish;

PERFECT!!  This is just what I was looking for; thanks so much, nellie_bly!!