According to dictionaries, the word international can be a noun or an adjective. As an adjective (describing word), it means involving two or more countries, or existing between countries. Examples: international mail, international food... As a noun (a person, place or thing) it is part of the name of a thing that is described by the adjective Examples: Rugby International, First International... In Hong Kong English it has an additional meaning both as a noun and adjective. Here, it also means upper-class. (The term upper class is not normally hyphenated. I do so here for the sake of clarity.)
Originally Hong Kong international schools were created to serve the needs of people from other countries who wanted to keep their children in the same curriculum as in their home country. Also since the children could not speak Cantonese, enrolling in a local school was not a possibility. However today most children in international schools are local Hong Kong Chinese. Their parents pay very high tuition to get their children a high-social-class education. Sometimes, but not always, international schools provide better education. Many local schools are better than some of the international schools. Many expatriates in Hong Kong are unable to afford the high cost of these schools. Today most of these international schools primarily serve local wealthy families.
Some businesses use the name international to mean upper-class. For example, the BP International Hotel is a very good quality hotel. It has high class decorations and service. The restaurant in the basement serves the best Hong Kong food. The hotel is owned by the Scout Association of Hong Kong. BP stands for the founder of the Scout Association, Baden Powell. Scouting is international, but this is a local upper-class hotel.
An expensive store or mall claiming to be international might sell expensive, upper-class goods from other countries. However, lower and middle-class stores also sell goods from other countries. That is common in Hong Kong. The real difference is that the expensive international store is upper-class. To be international in the usual sense, a business must have shops in two or more countries. A Mainland shop does not count as being international.
Protesters who ask for international standards seek improved political standards. They forget that when Hong Kong was ruled by the United Kingdom, it was a colonial dictatorship. They do not really want the political flaws that other countries have. The United States has an election committee called the electoral college; as a result most people's votes don't count mathematically. In parliamentary countries, such as the UK, Australia and Canada, the prime minister is not chosen by a national election. They are chosen by political parties making an internal members-only vote. I have participated in such an internal party vote, but most westerners have not. The public only gets to vote for their local representative. I could go on to more extreme examples. Internationally, democratic politics are usually flawed. What (some) protesters are asking for is an improvement.
Other words, such as world, are also used to make similar exaggerations. Some businesses in other places might use this form of exaggeration, but there it is the exception. In Hong Kong English this use is common place.
The usual definition for international is still used in Hong Kong, but it is supplemented with a secondary meaning. When people use the word, we need to think twice and check which meaning they intend.