So, I noticed that, at least on the Samsung Galaxy S3, when you exit a video(or movie) you're watching in the phone's media player, it starts the video over again next time you open it. But I figured out a work-around! All you have to do is place the Video Player widget on your homescreen, then open the video through the widget when you want to resume it. BOOM! Problem solved. Movies and videos will be much easier to enjoy now!

Here's a photo so you can see what the widget looks like (I use Go Launcher Ex with a 5 icon width, so if you're using a different launcher or have different settings it may be bigger or smaller):
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