Considering your requirements for low noise and a budget of $400, you should be looking at an electric guitar and possibly one that comes in a package deal. The package deal would include the guitar, amp, some picks, a strap, and a tuner and maybe some other things. Check around to see what is available. I really do recommend going to a guitar store so you can play different guitars and actually play the one you choose in the end. Make sure they allow you to play the one that is in the package because all guitars can be a little different.
I do not think that the term "semi-electric" applies to guitars. There are "semi-acoustic" guitars which have a small hollow in the body of the guitar or even those with piezo pickups and sound like an acoustic guitar when amplified. The last type of guitar would be something like found at this link:
http://www.jimlaabsmusic.com/guitars/acoustic-guitars/godin-a6-ultra-semi-gloss-semi-acoustic-electric-guitar-natural-cedar/prod_1401.html
If a guitar has the equipment to be amplified then it is an electric guitar in the sense that it uses some sort of electronic components to amplify and electrical signal to send to an amplification system. However, the term "electric guitar" is most commonly used to describe a solid body guitar with magnetic pickups.
There are also acoustic-electric guitars which are acoustic guitars with an electronic device used to change the vibrational energy into electrical energy as a signal to be amplified. The electronic device is integrated into the guitar. Then there are acoustic guitars that people use with soundhole pickups to change the string vibration into an electrical signal to be amplified. There are other configurations but those are the most common ones.
Again, it sounds like you are looking for an electric guitar with an amp with a headphone output. Audio input into the amp would also be nice for you so you can play with audio tracks but it is not necessary.