You almost answer your question yourself: The answer is that there isn't a single, standardized way of joining the stars. The reason for this, from a scientific point of view, is that whatever creature some pattern of stars makes you think of bears no scientific value. We don't learn anything about the Universe by drawing these lines. They're beautiful reminders of fascinating myths, but myths have different versions and origins, so e.g. Orion can be drawn as wielding a club or a sword in his right hand, and a shield or a bear hide or a bow in his left.
That doesn't mean that the patterns are useless: To categorize astronomical objects and phenomena, it is sometimes helpful to divide the sky into regions, and one way to do this is to follow the historical constellations. For this reason, the International Astronomical Union defined, in 1928, exact boundaries between 88 constellations, following the directions along and perpendicular to the celestial equator (measured through right ascension and declination, respectively).
Bright stars are then sometimes named by a Greek letter (in general, but not always, starting with α for the brightest, β for the next-brightest, etc.), plus a three letter abbreviation of their constellation. For instance the eastern-most star in the Big Dipper, which is in Ursa Major, is called "η UMa".
However, as with most objects in astronomy, η UMa has other names as well: In the Hipparcos Catalogue it's called HIP 67301, in the Henry Draper Catalogue it's HD 120315, and when you're stargazing it's called "Alkaid". And that's good because in 62,900 years it will have crossed the border to the neighboring constellation, Canes Venatici (if you can trust Stellarium's prediction of star position beard on the proper motion). So even this practice of dividing the sky into constellations is not that helpful after all, and is of course completely arbitrary.