Hmmmm… I guess you would be a sort of Software Engineer…!
At the end of the day though ‘Engineer’ is just a label – if you think you want to program PC games then whether or not you can call yourself an ‘engineer’ is kind of not important… Go for jobs which have good content (i.e. they do things you enjoy doing) rather than a good label…!
I remember at university our computer science department had a degree in Software engineering specialized in Game design and programming. I remember they had Xboxs and PlayStations in their labs and they were playing/designing games. I have always been jealous! So the answer is simply yes it can count as a form on engineering.
Yes – it can be part of software engineering. Games also have graphics designers, story designers…. The government is very keen on promoting this industry in the UK, so it’s probably a good profession to get into at the moment!