I always recommend updating your resume the day you start a new job. You may never give anyone that resume before it is next updated, but on your first day you know the dates, the company name, your job title and so on, and what better time to capture them than now? (It's also the perfect time to adjust the description of the job you just left - putting things into the past tense if they were present tense, for example, and making sure your job title and duties are complete.)
A year or two from now you may not remember if you started in September or October, or other details that are easy now. Your resume is a living document - put what you can on it now, and adjust it regularly as your job duties shift. That way there's never a big ugly update my resume task that's blocking you from looking for a new job should you want to.
For the specific resume you send to apply for a job that will start 8 months from now, include the job with start and end dates, the end date being in the future, and use the future tense along with verbs like "expect" or "plan". For example:
Sept-Dec 2015, Web Developer, BigCorp. Designed software for XYZ. Duties for the remainder of the term will include A, B, and C. We expect to deliver D and E.
I would not think it "looked bad" if your application for a temporary internship at my firm included a different temporary internship or project position that you started 8 months before my start date. I would think it looked bad if you were applying for an internship that started right now, since you'd be saying "I'll totally ditch these guys for you if you offer me something good." But that's not the situation you're in.