Trigger is a more generic term that can cover transmitters, receivers, transceivers, optical slaves, etc. It just means a device that can trigger the flash burst. A transmitter, very specifically means a radio transmitter that sends the triggering signal. And yes, a transceiver is a unit that can act as either transmitter or receiver.
Whether you need two units depends on whether there's a receiver built into the flash, and whether there's a transmitter built into the camera and what triggering system you want to use. For example, a D7000 has a built in CLS transmitter, and the SB-700 has a built in CLS receiver, so no radio units are required to use that flash off-camera with that body. That same D7000, however, used with a YN-560III would require one transmitter for the camera, because the YN-560III has a built-in RF-602/603/603II/605 receiver built into it, but the camera doesn't have a radio transmitter built-in. And if you were using RF-603II transceivers with the SB-700, you'd need two units: one to act as transmitter on the camera, and one to act as receiver on the SB-700.
You can use Phottix triggers on other brands of flashes, but how much function you can get might be limited to making sure that you get a brand-compatible set of Phottix triggers (i.e., Phottix triggers made for Canon/Nikon/Sony etc.) So, yes, you can trigger both your SB-700 and YN-560 with Phottix/Pixel/Yongnuo/PocketWizard/Radiopopper/etc. etc. triggers if you want.
Manual triggers that allow for remote power control are coming on the market now, however if you are not using TTL gear, then mixing-and-matching brands may not be an option. I.e., The Yongnuo YN-560-TX transmitter can control the power level of a YN-560III, but only a YN-560III (i.e., it could not control the power level on the SB-700. If you used the YN-622N TTL triggers, you could control the power on an SB-700 and the TTL-capable YN-568EX (Nikon flavor). But not a YN-560III. The Cactus V6 triggers can let you control power levels with Nikon/Canon/Pentax TTL-capable/compatible flashes, but not all-manual flashes. And the Godox FT-16S can only control the power on Godox manual flashes.
A manual-only flash is very very limited in what it can do and communicate through its flash foot, because the flash foot has only one pin: the sync signal. The reason same-brand gear (like the YN-560-TX and YN-560III) can have additional features other than remotely firing the flash is because the same company can make the transmitter and the built-in radio receiver--and add additional function through that communication. But that built-in radio receiver is only in that model of flash.
TTL gear, however, has the full complement of pins and signals to communicate the entire flash hotshoe electronic communication protocol, so triggers can then send/receive between those units most of the features you can access with the flash on the hotshoe.
Aside from all of this, you actually can set off both an SB-700 and a YN-560 model (any mark) without radio triggers or CLS by using the built-in dumb optical slave modes of both flashes. This type of slave is very simple: a sensor on the flash will trigger the flash to fire when it "sees" another flash burst. On the SB-700, this is called SU-4 mode, on a YN-560, it would be labelled S1. And all that can be communicated is the "fire!" signal. You'd have to set power on the flash units themselves, and you have to use the pop-up flash on your camera (assuming it has one), in manual mode, to command the remote flashes. You can dial the power down to "--", so that the commanding signal won't register at regular subject distances, but the pop-up will have to fire to trigger the remote flashes. It needs to be in master (and out of commander mode for CLS if it has that capability), so that TTL or wireless CLS preflashes don't trip the remote flashes before the exposure is taken.
Optical slaving (whether CLS or "dumb" optical), does have the drawbacks of range and line of sight limitations--particularly when used outside in bright sunlight (which is why radio triggers are so popular), but can work very well in studio-type conditions, and can be a great starting point to learn off-camera flash.