In short, they are both scams that fall heavily into the category of Not Even Wrong or cargo cult science.Tesla's work is sadly a great favourite of conspiracy theorists who do not have the technical background to adequately understand why these devices do not function as intended.
Your "Cold Jet" link should scream "I do not know what I am talking about" not only because of the broken, rambling sentences (although admittedly thats not exactly a rare quality amongst many actual professors) but also because of the over reliance of handwaving any problems away to just an "on board compressed air source"; without any mention of exactly how this on board compressed air source will itself be powered.
As to the tesla gas turbine itself as pictured, just because some guy can poorly model something in Solidworks and take a screenshot of it does not make it a viable design, nor does simply claiming "it has a high efficiency!" make it so.
The disc shaped turbine design apparently works on the basis of the boundary layer viscosity. This is a real effect and to say that boundary layers have been studied intensively is like saying that Everest is larger than a grain of sand.
In this sense the turbine works, very much like an automotive plate torque converter with air instead of hydraulic fluid. It may be an "efficient" design (although I have not been able to find any kind of source that gives any source of figures, accurate/reliable comparison or indeed methodology; and it has never been in any single one of my undergraduate or graduate textbooks) but the torque it is capable of producing is minimal due to fluid shear.
Any kind of small deviation in the surface will lead to turbulence which, on the scale of the disc separation will cause a complete destruction of the boundary layer on which this device depends; which makes this an incredibly temperamental device probably only useful for a single speed which is no good for any practical application (apart from any other flaws it has).
Note that I have concentrated solely on the turbine aspect of this - again, I have not been able to find anything remotely approaching a scientific analysis of this concept with applications as a tesla compressor. Simply having a bunch of discs revolving in close proximity does not a compressor make. You might manage to raise the static pressure of the fluid in close proximity to the rotating elements by a few Pascals solely because of the velocity effect, but you will not experience anything like the compression needed to sustain efficient combustion.
In both cases (compressor and turbine) the close spacing of the plates of the devices will paradoxically probably rob them of any claimed "efficiency" due to their very model of operation (fluid shear) and the boundary layer drag that results. This will also severely limit the mass flow rate through any such device; which in turn severely limits the practical application.
You are better off looking elsewhere. Even model radio control jet aircraft use a conventional miniature centrifugal compressor wheel taken out of a super/turbocharger; with an two-stage axial turbine to provide power.