"chapter" is the common word used for "Stift" and definitely found in connection with Augustinians. I don't think there is a special word for "Klosterstift".

Building on Veronika's answer. In Austria, Stift is usually translated as Abbey, but monastery is also used (see below).

ab·bey (āb'ē)
n. pl. ab·beys

1. A monastery supervised by an abbot.
2. A convent supervised by an abbess.
3. A church that is or once was part of a monastery or convent.

[Middle English, from Old French abaie, from Late Latin abbātia; see abbacy.] The American Heritage® Dictionary of the English Language, Fourth Edition

mon·as·ter·y (mŏn'ə-stěr'ē)
n. pl. mon·as·ter·ies

1. A community of persons, especially monks, bound by vows to a religious life and often living in partial or complete seclusion.
2. The dwelling place of such a community.
[Middle English monasterie, from Old French monastere, from Late Latin monastērium, from Late Greek monastērion, from Greek monazein, to live alone, from monos, alone; see men-4 in Indo-European roots.]
mon'as·te'ri·al (mŏn'ə-stîr'ē-əl, -stěr'-) adj.
The American Heritage® Dictionary of the English Language, Fourth Edition

If you look at the websites for Stift Melk, Stift Heiligenkreuz & Stift Klosterneuburg, you'll see some of the different uses. Melk & Heiligenkreuz use "abbey" to translate the name, but Melk for example sometimes uses "monastery" in the text when referring to history or the buildings. Stift Klosterneuburg is translated as Klosterneuburg Monastery, but I don't know enough about the monastery to know if there is an abbot or not.

http://www.stift-klosterneuburg.at/

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Note added at 6 hrs (2010-05-28 11:05:42 GMT)
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that's abbey, lowercase, of course

The biggest problem with one English equivalent for each term is the scope of time involved. Up to 1200 or so, a Kloster would probably have had an abbot or abbess as head, and could also be designated an abbey and not merely a monstery/nunnery. It was certainly the latter, as the medicants get their start in the early 1200s, but it could also be a priory.
Klöster that were founded by other Klöster as daughter-houses were ruled by priors/prioresses, so are correctly priories, not abbeys, but could also be called monasteries/nunneries.

Stiftskirche as a 'collegiate church' could work, but it is narrow definition of the term, implying only those foundations of canons/canonesses.
It is minimially a church endowed by a specific patron. It has a priest (and assistants), money provided for the upkeep of the building and the priest's wage (they've got to eat, too), but not necessarily a religious community.
Collegiate does not merely *imply* a relgious community, it *demands* one. This would have be a community of canons regular or canonesses.
Stiftskirche is also a term commonly used in English to refer to specific churches in German speaking areas, and might be a way out of your conundrum if it were glossed the first time it occurred. That you would have to check with your client. Otherwise, I find it to be context specific, and sometimes in reference to a church, a collegiate church, a private chapel, a priory church, or a monastic church. For example, St. Gallen is a Stiftskirche, and has always been a Benedictine monastery.

A monastery/nunnery (Kloster) could have daughter houses (Klosterstifte), which may have been ruled independently by an abbot/abbess (unlikely), but more likely by a prior/prioress, and are often called priories for that reason.
BTW, modern monks and nuns in German speaking areas refer to the 'Kloster' as only that part of the monastery that visitors may not enter (or visitors of a certain sex may not enter, the library at Einsiedeln), and that area varies according to order and individual house. (Just to muddy the waters.)

The inhabitants of monasteries swore 3 major oaths, obedience, chastity, poverty (stability is often added, meaning that monks/nuns should not leave their house unless under orders). The inhabitants of daughter houses founded by monastic orders swore the same.
Canons and canonesses swore obedience and chastity, but not povery. They were often lay religious, meaning that they entered the life as adults. Initially, all canons were ordained, which was the original distinction from monks, who were often not.
Canons often had individual sources of income, which meant that not only did the houses vary in wealth, individuals within houses often had widely different lifestyles. (Not all monks were ordained, but that is a difference defined more by order and time frame).
In the 12thC, there is a broad movement to regularize the canons/canonesses, which is when the Augustinian Rule is broadly imposed (it had been followed by a lot of them, but not all).

In your first example, the Stift is inhabited by canons/canonesses, which means that it is a House of Canons/Canonesses. If this is the distinction that the author is using throughout the text, then I would go with this. It is also supported by your fifth and sixth examples

However, a Stift could be inhabited by monks/nuns/canons/canonesses, be ruled by an abbot/abbess/prior/prioress.
It means a religious foundation based on a donation of land/tithes/rents, usually sponsored by one patron (bishop, layman/laywoman, other religious order), who is honored by special privileges in the house (accomodation, right of burial, prayers).
A Stift, founded by a bishop, was often merely a church where mass was celebrated (requiring a priest and assistants, but no monks et al), distinct from the parish church where the local inhabitants worshipped.
While some monasteries served as parish churches as well (Shrewsbury comes to mind) they had to have a parish priest to administer to the locals, while the monks had their own priests for their mass. Parish churches only rarely became a monastic church. Stifte, otoh, esp those endowed by bishops or powerful laypatrons, could grow beyond the initial church into monasteries/nunneries/houses of canons/canonesses.

In your second example, Stift merely refers to a non-Benedictine foundation, which offers the possibility for priories, other monastic orders (Cistercians, Cluniacs, etc), a house of friars (Dominicans or Franciscans), a hospital, a church. Without more context, I can't tell if your author is singling out the Benedictines because it is a history of the order, or is tracing a specific set of religious foundations among which Benediticine houses form a large percentage.

As Veronika pointed out, and to which you have agreed, a Klosterstifter is the founder of a monastery/nunnery/priory, often called the patron (not to be confused with the patron saint, also known as the patron, just to be confusing).

In some of these contexts, "Stift" simply means "foundation". This may refer to a newly established religious house of an existing order, or a completely new religious order.

The latter seems to be the context for this sentence:
"Neu gegründete Stifte folgen nun zumeist der Augustinus-Regel."

I'd render it as:
"Most newly established foundations follow the Rule of St. Augustine."

Similarly, a "Klosterstifter" is a "founder of the monastery". In some cases, a monastic founder may have been a benefactor rather than a monk or nun.