Is this an s-trap and can I vent it in the cabinet?
The definition of an s-trap is where:


There's a drop in the pipe downstream of the trap weir (or outlet), and
That drop occurs before the vent connection


You hit both points, so yes, it's an s-trap. However, if it's not vented properly anyway (as your diagram seems to indicate), it's all moot. Such a situation always acts as an s-trap.

The solution is simply to extend your tailpiece, assuming you end up within maximum drop (might be 18"). 

Whether you can retrofit a local vent depends on your community's code authority. In the U.S. air-admittance valves and other internal vents are no longer legal, generally speaking. You're required to vent through the roof. 

In no case can you simply add an open tube, as that would allow sewer gas to escape and defeat the purpose of the trap, and it needs to be the size of the drain pipe anyway or it won't vent adequately.

There are other venting retrofit options and exceptions, but we'd need to know more about your plumbing.