Based upon the above information, you are correct in saying that something is wrong with your understanding of the word "there." Depending upon the dictionary you use and upon the function of "there" in a sentence--whether it is defined, as in Dictionary.com, as having a syntactical function of a pronoun, a noun, an adverb, an adjective, or an interjection--"there" has many, many meanings. The definition you site for “there” can be found on Dictionary.com under the heading of Interjection: "(used to express satisfaction, relief, encouragement, approval, consolation, etc.): example: 'There! It's done.'" This definition of “there” as an interjection is confirmed in Longman Dictionary of Contemporary English:
1 spoken used to express satisfaction that you have been proved right or that you have done what you intended to do:
There! I've done it! I've resigned.
There, what did I tell you? I knew it wouldn't work.
Interjections are also called exclamations as in Cambridge Advanced Learner Dictionary. On the other hand, this use of "there" is defined by Oxford Advanced Learner Dictionary as an adverb "used to attract somebody's attention: Hello, there! You there! There you are!" Note that “there” in “Hello, there!” is not related to direction, location, or place: it is an interjection and exclamation, therefore the inclusion of "(out)," which you presume to be omitted, is an error in syntactical understanding.
One thing all these dictionaries agree upon is that the use of "there" as an interjection or exclamatory adverb for attracting attention has no relationship with "there" when used to indicate location, as is done with the contrasting adverbs of place, direction, and location "here" and "there." By conflating (i.e., putting two separate things together that should be left apart) these meanings (i.e., those of interjection and place), you attempt to understand one meaning in terms of another meaning when they in fact can be considered to be different words: "there" the interjection, exclamation, or adverb for attracting attention is not the same as "there" the adverb of place, location, direction.
So, while you are correct in understanding
There, I've made it work at last.
There, now I can have some peace!
There! Is everybody happy now?
as interjections/exclamations/adverbs attracting attention that express "satisfaction, relief, encouragement, approval, consolation, etc," you are not correct in thinking this sort of "there" is related to "there" the adverb of place/location/direction that is paired with the adverb "here" similarly used for place/location/direction. As previously, both "there" and "here" have many more meanings than the two under discussion. Your clearest understanding will probably come from examining Oxford and Cambridge dictionaries for "there" and "here." Longman can then be helpful in understanding the syntactical uses of each word.