As you can see, there are many terms that were used in the past for the position of slubber. The job title depended on when in the process of producing the woven fabric the job was performed and what tools were used.

ÉPETISSEUR Ouvrier enlevant à la main les irrégularités d'un tissu.
ÉPINCEUR Tailleur de pierre.Paveur.***Ouvrier enlevant, à l'aide de pinces, les mauvais fils des étoffes fraîchement tissés.***
RATTACHEUR Ouvrier d'une filature qui était chargé de nouer les mèches ou les fils cassés sur les métiers.[Not a slubber. Actually it is his work that the slubber had to fix.]
http://50864.aceboard.net/50864-742-691-0-.htm

Épetisseur - Ouvrier enlevant à la main les irrégularités d'un tissu.
Éplucheur - voir Arracheur.
Épinceur - voir Énoueur.
Épincheleur - voir Énoueur.
Époutisseur - voir Énoueur
Énoueur - Ouvrier qui ôte avec de petites pinces les nœuds (énoue), pailles ou bourrons à la surface des étoffes.
http://www.vieuxmetiers.org/lettre_e.htm
Arracheur - Dans la chapelerie, ouvrier qui ôte avec des pinces le jarre de dessus les peaux de castor.
http://www.vieuxmetiers.org/lettre_a.htm#arracheur


Baqueteur - ouvrier tisserand tirant les fils avec un outil nommé "baguette".
Ebroueur - ouvrier enlevant les noeuds des toiles ; casseur de noix.
Enoveur - ouvrier enlevant les irrégularités d'un tissu (région laonnaise).
Epetisseur - ouvrier enlevant les irrégularités d'un tissu (région laonnaise).
Epinceleur - ouvrier enlevant avec des pinces les mauvais fils des étoffes fraîchement tissées.
Epoutier, époutilleur de drap - équivalent d'épinceleur.
http://www.cevenols.com/vieuxmetiers.htm

Machine related info for the glossary
épurateur de fil: appareil purgeur (m) - yarn cleaner: yarn clearer: slub catcher
http://www.centexbel.be/fr/lexicon_E.htm

see also the vocabulary that I posted at http://www.proz.com/kudoz/1209324
which included:
épuration nf (fil.) slub catching
épuration électronique electronic slub catching


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Note added at 7 hrs 47 mins (2006-01-04 19:02:28 GMT)
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Here is the one for cleaning bales of cotton:
éplucheur de balles: éplucheuse de balles (m) - bale plucker: bale picker
http://www.centexbel.be/fr/lexicon_E.htm

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Note added at 10 hrs 3 mins (2006-01-04 21:19:08 GMT)
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From what I've seen, the slubber's job wasn't to clean cotton and this is where I think our confusion has come from. There is a difference between the process of cleaning fibers or slubs from textiles and the process of drawing down the fibers from thick slivers to medium slubs to thin threads.

Cotton is ginned before it is brought to the textile mill for spinning. At the mill, the bale is broken open, beaten and fluffed and formed into laps before it is handled by the slubber. I haven't found anything to support the text that you have which shows slubbers as actually cleaning the cotton.

Cotton Manufacturing
The scutcher.-If there is considerable dirt in the cotton it is generally run through another series of machines known as scutclaers. In the scutchers the cotton lap gets a great deal of beating and shaking which removes the foreign matter.

The cleaner the cotton is at the start, the less beating or scutching it needs. This is an important matter, for beating is sure to weaken a certain amount of the fiber, and the more beating it gets, the weaker will be the final product, the yarn
[see this site for a complete description of the processes involved]
http://www.oldandsold.com/articles04/textiles6.shtml

SLUB: A thick place in a strand of YARN caused by improper SPINNING.[Notice the difference between slub and slubber]
SLUBBER: A machine which DRAWS OUT strands of SLIVER and twists them together loosely in order to give the strands (now ROVING) sufficient strength to withstand subsequent operations.
SLUBBING: The process of drawing out and loosely twisting SLIVER to make ROVING.
BREAKER PICKER: The first of the two units of older style PICKER MACHINES. In this unit the raw cotton is partially cleaned by beating and fluffing and then fed into a FINISHER PICKER.
FINISHER PICKER: The second of the two units of older style PICKER MACHINES. This unit receives partially cleaned cotton in the form of LAP from a BREAKER PICKER and completes the cleaning and fluffing process.
http://www.bitwisegifts.com/glencoenc/library/doc4.htm
SPINNING ROOM
Drawing frame $4.50
Slubber hands
http://www.ibiblio.org/sohp/laf/factory.html

Next the cotton is fed into a "lapper," which continues the work of untangling the fibers, removing impurities, including broken ends or short lint. It is delivered in the form of "laps," which are sheets of batting of loose texture 36 to 45 inches wide and usually 48 yards long, weighing from 10 to 18 ounces the yard...
The sliver is now fed to the "slubber," which reduces the thickness and imparts a slight twist. The product is now "slubbing" and is wound upon large bobbins to be ready for the next process.
http://www.hampsteadchamber.com/A Southern Primer/cotton.htm

SPINNING ROOM [a slubber worked in the spinning room according to this site]
Drawing frame $4.50
Slubber hands
http://www.ibiblio.org/sohp/laf/factory.html

Cotonneur - ouvrier travaillant le coton.
http://www.cevenols.com/vieuxmetiers.htm

ouvreur/ouvreuse de balles
nettoyeur/nettoyeuse de machine textile
nettoyeur/nettoyeuse de machines textiles
nettoyeur/nettoyeuse de matériel
nettoyeur/nettoyeuse de matériel de fabrication
http://jobfutures.ca/cnp/titres/961x.shtml