Thermodynamic vs kinetic basicity
I have worked on a computational project where I had to study the sites of deprotonation in this molecule:



When deciding between mechanisms I had to consider both thermodynamic and kinetic. For example, I had this case: deprotonations of two different hydrogens(bond with N Blu atoms in the picture)were similarly thermodynamically favored (similar free Gibbs energy of reaction) but one H was towards the inside of the ring and the other pointed towards the outside, therefore, because kinetically favored, the mechanism proposed was the one with the external H removed.

Your questions:


  What is the exact distinction between these two terms?


A Thermodynamically favored base means that the Gibbs energy of the rection where an H+ binds to the protonation site(eg.lone pair of the N) is very negative.
However, if this protonation site is difficult to access, then this reaction is very slow or even impossible and a less thermodynamically favored protonation reaction may occur instead(if the molecule has more than one protonation sites or other bases are present). 


  Is thermodynamic basicity the one mentioned in common basicity defintions? (arrhenius, lewis, bronsted-lowry)


yes, and the kinetic effect is evident only in limited numbers of cases.


  Why are they measured differently?


To make it easy and avoid competitive reaction considerations, suppose you have only one type of protonation site. If you want to know how easy is for the proton to access that site, you have to determine the speed of the reaction(dynamic experiment). If you want to know only about thermodynamic(energies involved) then you let the reaction go to equilibrium and just measure the pKa. If the deprotonation site is very inaccessible then this thermodynamic experiment may take very long because you have to wait it reaches the equilibrium. In this case, if it takes more than your experiment time scales than you will measure an apparent pKa that is lower than expected because your reaction has not reached the equilibrium yet. This apparent lower pKa is due to the kinetic effect.


  Are they denoted differently, do they have different units and is there some general mathematical formula for each that would help me get deeper understanding and distinction between these two terms?


I don't know if there are other ways to express this other than kinetic and thermodynamic acid/base. 
Thermodynamic: you wait for the reaction to reach the equilibrium and using the $\Delta G$ of reaction determine the pKa.
$ΔG = -RT\ln K_\mathrm{a}$
 $pKa = -log(K_\mathrm{a})$
 In case of a difficult to access protonation site you may not wait for enough and measure a lower pKa but this is an apparent pKa because the definition of it is purely thermodynamic. 


  Also, how are these two mesaures of basicity related to gas phase proton affinity and intrinsic basicity and pKb?


Not expert of these techniques, I worked only theoretically but as usual, the solvent affects the value of the pka. The most stable conformation in the gas phase may not be the most stable in solution anymore and there's the solvation energy contribution to consider when you determine the Gibbs energy of the protonation reaction. Therefore, the measurements in solution may not be comparable with the gas phase.

Finally, I can give you some example of very evident kinetic effect:
 In the molecule below the lone pair of the nitrogen is difficult to access for an H3O+ molecule and the pKa measured(not the thermodynamic pKa) will be different depending on how much you wait during the titration experiment. I don't know the timescales for this molecule. 


or you can have a base trapped in a fullerene molecule:



This is the most extreme case with an infinite equilibrium time. The pKa of the molecule trapped is still the same but you will experimentally measure the pKa of your solvent and conclude it is not basic.