First, read the article linked to by denis, which gives yet a third and more sophisticated formula for percentile, which is one type of descriptive statistic.
Second, the purpose of descriptive statistics is to reduce a large and apparently confusing mass of numbers to a few easily comprehended numbers. These simplifying numbers virtually always lose information. The hope is that the information lost is worth less than the gain in understanding. The statistician's duty is to pick the type or types of simplified number that provide the maximum degree of understanding.
Third, the "better behaved" the data are, the less information is lost by relying on descriptive statisitics.
Fourth, by "well behaved" data, we mean that a graph of their frequency results in a relatively simple curve. The intuitive idea is that a few simple numbers are sufficient to describe a simple curve. By "lumpy," I meant that the data are heavily concentrated in a small number of values rather than being spread across a large number of values.
The various formulas discussed attempt to improve the reliability of the percentile as an informative summarizing number when the data are few, not well behaved, or quite lumpy. When the data are ample, well behaved, and not lumpy, all three formulas give virtually the same result. Because mathematicians have not agreed on them, it should be obvious that none of them is conclusively "right." Consequently, I at least cannot give you an answer to your question about 0.5 except that it is easy to work with and takes a neutral position between counting all of the instances of one score and none of the instances. It is a reasonable adjustment.
In my opinion, percentiles simply are not very good summarizing numbers whenever the number of data points is small, the data are not well behaved, or the data are lumpy. I'd presume when someone reputable gives data in percentiles that the data are numerous, well behaved, and not lumpy, and the question of which formula was used has virtually no effect on the numbers given. It approximately gives the percentage of total scores that are below the indicated score, and 99 minus the percentile approximately gives the percentage of total scores that are above the indicated score. So a score in the 70th percentile means approximately 70% of the scores were lower, and approximately 29% were higher. If those statements are quite misleasing, then someone reputable will probably not use percentiles as a descriptor.