I'm going to answer with a short overview of the history of the study of clothing in the Middle Ages (including some sources to check out at each step) and end up discussing your specific question.
I'm not surprised that there is confusion around the naming of different garments. This is partially due to the bifurcated evolution of the history of clothing.

The field of costume history has its roots in two fields: art history, and theatre. Because of the number of extant garments from the medieval period is quite small (although significantly larger than it was the Victorian era, where we see most of the initial works dedicated to a history of costume), the early works on costume history tended to focus on paintings, illumination, and sculpture. Art historians were less concerned with practical matters around clothing and more on either the symbolism or meaning of clothing in artwork, or, eventually, in the items themselves as examples of art (e.g. embroidered garments). However, it did not take long before another stream emerged from this, which was focused on more practical matters--that is, how to make clothing for the theatre (and eventually, for the cinema) that was "correct for the period." One of the first works to emerge from this was A History of Costume, credited to Carl Kohler (who died in 1876) but "edited and augmented" by Emma von Sichart in 1928. What Kohler did was to look at extant garments and to produce workable patterns from his measurements--at least, that's what he purported to do. What he in fact mostly did for the medieval period (and for earlier periods--any time he did not have an extant garment) was to draft his own patterns based on his observations of pictorial or sculptural sources--with very mixed results.

The next generation of writers on costume began to tighten their focus and to correctly identify the evolution of style based on direct, close examination of illumination, painting, and sculpture, but with more study of the actual terms used for garments in literature and other written sources. These sources (for example, C. Willet and Phillis Cunnington's Handbook of English Mediaeval Costume and Mary G. Houston's Medieval Costume in England and France) date from the 1930s through to the 1970s, and are still useful if you are looking for depictions of costume from particular periods compiled together in one place. You will also begin to learn some of the common terms for items of clothing and when they were used. For instance, regarding the terms you mention-- cote/cotte and tunic are fairly generic terms for an outer garment (as opposed to an undergarment) covering at least the upper body, for both males and females, while bliaut is a garment made from luxury fabric (likely silk), which may have fur lining or trim and/or gold ornamentation; this term is generally only used in the 12th and early 13th century. Surcotes first show up as a clothing item in the 13th century (they're used with armor much earlier); the term denotes a sleeveless outer garment worn over another garment. And the "cote-hardie" is a nickname for a particular type of cote worn in the 14th century by both men and women. Some of these also contain information on construction, which is not always reliable.

At the same time, the main sources for those interested in cut and construction of medieval clothing continued to be books targeted at a theatrical audience. Iris Brooke's Medieval Theatre Costumes is an example of this type of work, where the focus is less on strict accuracy and more on verisimilitude. These works are generally not good sources for understanding the correct terms for garments.

In more recent years, the study of medieval clothing and textiles has diversified. More and more evidence is being brought to bear from archaeology and studies of material culture (such as weaving, dyeing, leatherwork, embroidery, etc.) , as well as from literature, from documents such as wills, inventories, and household accounts. We now have more specialized works at a high academic standard, such as Gale Owen-Crocker's Dress in Anglo-Saxon England and Thor Ewing's Viking Clothing. There is better access to what extant garments do exist through sources such as Clothing the Past by Elizabeth Costsworth and Gale Owen-Crocker. Many individual studies of topics in the field of medieval clothing and textiles can be found in the yearly journal Medieval Clothing and Textiles (since you're interested in the 12th century, Monica Wright's article on the bliaut in vol. 14 may be of interest.) And spurred on by interest by the living history community, there are now even well-informed sources for the construction of medieval clothing, such as The Medieval Tailor's Assistant by Sarah Thursfield.

Bringing this back around to your specific question, if you're thinking more the middle of the 12th century, I highly recommend Nicole D. Smith's chapter on Marie de France in her book Sartorial Strategies. Marie was a courtly poet, born in what we now know as France, who probably wrote for the English court (but with Eleanor of Aquitaine on the throne of England, the two were closely associated). She's one of the poets who specifically mentions articles of clothing, including the bliaut and the chainse/chemise worn underneath. Her description, along with the pictorial and sculptural evidence, should hopefully give you what you need to both use the correct words and to describe the look of the clothing. For lots of depictions of the bliaut--which is not (contrary to popular opinion) just a French thing--this is a good article that also goes into construction.

If you want to go earlier, things may be a little more difficult, but the aforementioned Dress in Anglo-Saxon England may be helpful, as there would be continuity between the clothing styles of England and Normandy, especially as portrayed in the Bayeux Tapestry, which was completed in the 1170s.

You've gotten a fantastic response about the history of medieval clothing, but I thought I would tackle your question in a more nuts-and-bolts way. (As a fellow author of historical fiction, I'm very sympathetic to the need to find definitions and explanations!)

The first basic part of the answer is that there was not an inherent distinction between construction of court dress and everyday dress in the High Middle Ages. This is really something that would come about in the Early Modern and modern periods, where a man might have a court suit made along different style lines or of different fabrics than what he wore everyday, or a woman might be required to wear a gown made with a train and low neckline, and it's a result of developments made to the way clothing was constructed in the Late Middle Ages.

Following on from European antiquity, the common upper-body garment was a tunic made with out (or with minimal) shaping to the body and sleeves. I'll quote from an earlier answer of mine regarding this name:

We typically call it a "tunic", because it's easier to discuss the history of material culture when we agree on some base terms that can be applied to more than one setting - shoe vs. sandal vs. boot, for instance, each mean something different, and we can apply them to items in cultures where those English terms are/were not natively used to cut down on explanations. "Sandal" works perfectly well, and is simpler than "a foot-garment that is mostly open and less-structured". Likewise, we all understand "tunic" to mean an upper-body garment of varying lengths that is unfitted and made of rectangular pieces. That being said, words that translate simply to "tunic" could be used in the Middle Ages to describe more fitted garments.

Tunica was used to refer to this garment, but so were other words, and what is important is that there seems to have been no interest in making it fit closely to the body around the beginning of your period. Men's and women's tunics effectively only differed in length, and the value of a tunic was simply a function of the expense of the material in it or used to embellish it.

This began to change in late tenth century France. To quote again from that other answer:

The interest in fitted clothing, "well-cut through the body," first popped up in late tenth century France, where the wealthy, fashionable, and young began to wear the bliaut - a gown/tunic of expensive fabric, made very tight in the waist, so that it created horizontal wrinkles as it pulled across the body. The body of the ordinary bliaut was cut with one length of fabric from shoulder to hem, shaped at the sides. A variation was the bliaut gironé, made with a waistline seam so that a fuller skirt could be pleated to the tight bodice.^1 (There was also at this time a similar garment called a chainse - a tightly fitted linen or hemp gown. One could wear either a chainse or bliaut over the unfitted undergarment, or could wear a chainse with a bliaut over it that was cut to display parts of the chainse such as the embroidered neckline or hem.)

(If you want to read this earlier comment in full, you can find it here.)

So now I'm going to contradict myself. While the idea of distinctive court vs. informal dress wouldn't become firmly established as a lasting divide until several centuries on, Christina Frieder Waugh believes that the bliaut would transition around the middle of the twelfth century (at least in France) into a formal dress for court rather than something to wear regularly. At that point, the elite returned to the common loose tunic until the beginning of the fourteenth century, when we see the rise of more complex methods of tailoring and fitting garments to the body, the regular use of fastenings and lacing, etc. But that's beyond the scope of what you're looking for.

1. Construction before roughly 1600 always involves a certain amount of speculation, and the amount of speculation increases the farther back you go. So 1100? You have a lot of experimentation, a lot of discussion as to what's artistic styling vs. what's realistic representation of the fabric, and so on. The vast majority of the discussion and experimentation is happening among reenactors, so blogs are not necessarily a bad source on this. I would not worry too much about trying to describe your characters' bliauts in great detail beyond tightness, length, big sleeves, and fineness of the fabric.

I feel like in need to raise this subject again, even if both answers given by u/Herissony_DSCH5 and u/mimicofmodes were highly satisfying.

Please note that medieval clothing is enthralled in a “dictionary paradox”, all the more questioned by synchrony and diachrony concerns. The term “costume” is in itself diachronic: what would be called a “costume” then? As far as I know for the Middle Ages, nothing. “Costume” refers to “custom” (tradition, habits). So, the question at heart of it is how do we know that a particular form refers to a particular term? As an historian, I think that nothing supports the definitions, and this came mostly from the XIXth century when scholars blended art history with archeology and tried to link or match texts with sculptures, illuminations and stained glasses (which are on their own limited by material supports: pigments, colors, techniques…). To translate directly art into words would be like taking a Magritte or a Hooper to define what was XXth century clothing, there are prisms and layers from the paintings to their sources.

From what I’ve encountered there are very few and precise depictions of clothing and even rarer matches with archeological evidences.

Now to be fair, that would probably enrage most of reenactors, but most of their sources either come directly from such XIXth century historians or illuminations. So, I wouldn’t be that self-assured when it comes to distinguish a bliaut from a cote or a chainse or a chemise in various illuminations. But, don’t get me wrong, reenactors are very precious when experimental archeology comes into play.

Now, something that I’m certain of is that XIIIth century clothes were “in layers” and that during the XIVth century French clothing started to “shrink” and became “slimmer” sized (as early as the 1320’s mostly during the 1340’s). This is on account of a growing north/south trade and italian mercenaries who fought during the early Hundred Years War. Such clothing was frowned upon at first but then were used with the approval of court etiquette. So contrary to Nobert Elias, I don’t think there’s a definitive “Verhöflichung” (“courtization”) during that particular period (apparently the term wasn’t translated in English), meaning the society as a whole wasn't influenced in manners and customs by the nobility, but the other way round.

At last, for the XIIth century, Orderic Vital gave a depiction of Foulques IV’s shoes and young men who followed his trend. Of course, Orderic Vital didn’t like it all but in context, this depiction could have a political meaning (Foulques IV was an angevin and Orderic Vital was an Anglo-Norman priest).

BARTHES Roland, The Fashion System, Berkeleyn University of California Press, 1967 [o.e. Le Système de la mode, Paris, Editions du Seuil, 1967]

ELIAS Norbert, The Court Society, New York, Pantheon Books, 1983 [o.e. Die höfische Gesellschaft, 1969].

CARDON Dominique, La Draperie au Moyen âge : essor d’une grande industrie européenne, Paris, Éditions du CNRS, 1999.

CHIBNALL Marjorie, The Ecclesiastical history of Orderic Vital, Vol. III, Oxford, Oxford Clarendon Press, 1969-1980 [text in latin, p. 323].

COLLECTIVE, Le vêtement. Histoire, archéologie et symbolique vestimentaires au Moyen Age, Paris, Cahiers du Léopard d’Or n°1, 1989.

MADOU Mireille, Le Costume civil, Turnhout, Brepols, collection “Typologie des Sources du Moyen-Âge Occidental”, fasc. 47, 1986.

NORRIS Herbert, Medieval Costume and Fashion, Mineola (NY), Dover Publications, 1998 [o.e. Costume & Fashion Vol II: Senlac to Bosworth 1066-1485, London, J.M. Dent and Sons Ltd., 1927] [read on line]

PASTOUREAU Michel, Une histoire symbolique du Moyen Âge occidental, Paris, Editions du Seuil, 2004.

–, Jésus chez le teinturier. Couleurs et teintures dans l'Occident médiéval, Paris, Le Léopard d'Or, 1998.

–, The Devil's Cloth: A History of Stripes, New York, Columbia University Press, 2001 [o.e. L'étoffe du diable: une histoire des rayures et des tissus rayés, Paris, Editions du Seuil, 1991]

VIOLLET-LE-DUC Eugène, Dictionnaire raisonné du mobilier français de l'époque carolingienne à la Renaissance, Vol. III & IV, Paris, Librairie Gründ et Maguet, 1858-1870.

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