These are just common expressions where the longer form is abbreviated for convenience and reasons of space and presentation. English is very flexible in this respect.

As Paul pointed out, 'English people' are, more or less, only the 50 million or so inhabitants of 'England', which is the area of the British Isles south of the Scottish border and east of the Welsh border. So we'll presume that you mean 'English-speaking people', which includes many hundreds of millions more people, not only in the UK, but also worldwide. And why do we say 'Game over' or 'Download complete'? Because it's a form of "shorthand", used for a quick message flashed up on a screen. It's not a full sentence. The full form is, for example 'Your game is over' or 'This download is complete'. So, you aren't wrong at all. The simple fact is that these are not full sentences, but abbreviations. In titles and messages, it is normal to miss out little grammatical words such as articles, pronouns and auxiliary verbs. For example, if you are writing a quick note to someone to say that you're going out for a short while, you might write 'Gone to bank', rather than the complete form 'I have gone to the bank'. It's simply a form of shorthand for speed and efficiency.

Peter hit the nail in the head when he said when he said it's like a newspaper headline. News headlines, signs, ads and notices often use abbreviated language to inform the reader of something. They're not full sentences and no one would actually speak like that. For example, no one would ever say "download complete". They'd say "The download has been completed". However, "download complete" is very common on computers to inform the downloader that the download has been completed. Like Peter said, this is done to save space. Strictly speaking it isn't 'correct' English, but it's used so much it's not disputed. You can see the rules here: http://www.englishlessonsbrighton.co.uk/8-grammar-rules-writing-newspaper-headlines/

"Page Not Found" is one of them, I think.

Why do English speakers say... 'English people' means people from England, not people who speak English.