Mine too will be a partial answer.

I'll include data that I can quickly summarize and provide any other useful insights (such as any formulae) I can think of.  I'm not familiar with the website so I can't do a comparison / critique but plan to investigate it in the future.

<h2>Star Generation</h2>
Depending upon the response, I might put more of what I have up here.  But this took me the last 4 hours to put together.

<h3>Assumptions</h3>
In my own notes, I do not identify the assumptions I made - I simply made them, move on, and forgot them.  I'll try to document them as I go through my materials so some of my answers don't seem so mysteriously disconnected from reality.

 - A random distribution of stars - the reality is the type of star you
   see will depend heavily upon where you look.  My investigation
   assumed a jump point interstellar drive system and that the
   connections were random (it made life easier).
 - I've extended the spectral classifications to make my life easier.
   <A href="https://en.wikipedia.org/wiki/Stellar_classification#Modern_classification">Conventional spectral classes include I, II, III, IV, V, VI</a>.  I've added
 VI, VII, VIII, IX, X, and XI as shown in the table below.

$$\begin{array}{|c|c|c|}
\hline \text{Luminosity Class} & \text{Description} & \text{Origin of Class} \\
\hline \text{I} & \text{Supergiant} & \text{Conventional Classification} \\
\hline \text{II} & \text{Brightgiant} & \text{Conventional Classification} \\
\hline \text{III} & \text{Giant} & \text{Conventional Classification} \\
\hline \text{IV} & \text{Subgiant} & \text{Conventional Classification} \\
\hline \text{V} & \text{Dwarf} & \text{Conventional Classification} \\
\hline \text{VI} & \text{Subdwarf} & \text{Conventional Classification} \\
\hline \text{VII} & \text{White Dwarf (stellar remnant)} & \text{Conventional Classification} \\
\hline \text{VIII} & \text{Neutron Star} & \text{My extension} \\
\hline \text{IX} & \text{Blackhole} & \text{My extension} \\
\hline \text{X} & \text{Nova, Supernova, other} & \text{My extension} \\
\hline \text{XI} & \text{Hypernova} & \text{My extension} \\
\hline 
\end{array}$$


<h3>Frequency of stellar objects</h3>
I downloaded and analyzed several star catalogs and determined star probabilities using that information.  My analysis revealed the patterns seen below.  Be aware that the numbers for the stellar objects with a Luminosity Class above VI (subdwarf) is estimated by me and probably bears little resemblance to reality.

I use type and class to categorized stars and also use the H-R (Hertzsprung-Russell) diagram.  The table provided below was developed primarily by 


$$\begin{array}{|c|c|c|c|c|}
\hline \text{Lum Class} & \text{Spect Class} & \text{Description} & \text{Probability} & \text{Cumulative} \\
\hline \text{I} & \text{M+} & \text{Supergiant} & \text{2.45%} & \text{2.45%} \\
\hline \text{II} & \text{M+} & \text{Brightgiant} & \text{0.97%} & \text{3.42%} \\
\hline \text{III} & \text{M+} & \text{Giant} & \text{10.14%} & \text{13.56%} \\
\hline \text{IV} & \text{M+} & \text{Subgiant} & \text{4.58%} & \text{18.14%} \\
\hline \text{V} & \text{O} & \text{Ultraviolet Dwarf} & \text{0.16%} & \text{18.30%} \\
\hline \text{V} & \text{B} & \text{Blue Dwarf} & \text{1.84%} & \text{20.14%} \\
\hline \text{V} & \text{A} & \text{Blue-white Dwarf} & \text{1.70%} & \text{21.84%} \\
\hline \text{V} & \text{F} & \text{White Dwarf} & \text{2.29%} & \text{24.13%} \\
\hline \text{V} & \text{G} & \text{Yellow Dwarf} & \text{2.90%} & \text{27.03%} \\
\hline \text{V} & \text{K} & \text{Orange Dwarf} & \text{10.69%} & \text{37.72%} \\
\hline \text{V} & \text{M} & \text{Red Dwarf} & \text{53.01%} & \text{90.73%} \\
\hline \text{V} & \text{L} & \text{Infrared Dwarf} & \text{6.20%} & \text{96.93%} \\
\hline \text{VI} & \text{M - G} & \text{Subdwarf} & \text{0.24%} & \text{97.17%} \\
\hline \text{VII} & \text{G+} & \text{White Dwarf(stellar remnant)} & \text{2.40%} & \text{99.57%} \\
\hline \text{VIII} & \text{F+} & \text{Neutron star} & \text{0.31%} & \text{99.88%} \\
\hline \text{IX} & \text{N/A} & \text{Blackhole} & \text{0.10%} & \text{99.98%} \\
\hline \text{X} & \text{N/A} & \text{Supernova, Nova, other} & \text{0.019%} & \text{99.999%} \\
\hline \text{XI} & \text{N/A} & \text{Hypernova} & \text{0.001%} & \text{100%} \\
\hline
\end{array}$$

 - M+ - Object can have a spectral class of M - O
 - K+ - Object can have a spectral class of K - O
 - G+ - Object can have a spectral class of G - O
 - F+ - Object can have a spectral class of F - O
 - M - G - Object can have a spectral class of M - G

<h3>Star Generation</h3>
To generate your very own star, you may use the following steps:

 1. Generate a random number ($N_{random}$) between 0 & 1.
 2. Apply to the table above.
 3. If the number falls outside the range of Luminosity Class V - main sequence
    dwarf (e.g. $0.1814 > N_{random} > 0.7900$), you are done.
 4. Otherwise, $N = N_{random} - 0.1814$
 5. $M_{star} = \frac{C_1}{C_2 + C_3 \cdot N}$ in Solar Masses
 6. $D_{star} = M_{star}^{2/3}$ (assumes main sequence stars have roughly
    the same density) in solar diameters
 7. $T_{star} = 380 + 5500 \cdot \sqrt{M_{star}}$ in K
 8. $L_{star} = \frac{4 \cdot \pi \cdot \left(D_{star} \cdot 7 \cdot 10^8 \right)^2 \cdot T_{star}^4 \cdot \sigma}{L_{\odot}}$ in Sols
 9. Maximum age $T_{star} = \frac{1.15 \cdot M_{star}}{L_{star}} \cdot 10^{10}$ in years
 10. The middle of the Goldilocks zone $r_{planet} = \sqrt{L_{star}}$ in AU

The constants

 - $C_1 = 0.0171$
 - $C_2 = 0.000143$
 - $C_3 = 0.271$
 - $\sigma = 5.67 \cdot 10^{-8}$ - Boltzmann's Constant

<a href="https://en.wikipedia.org/wiki/Mass%E2%80%93luminosity_relation">Wikipedia has a more accurate Luminosity calculation</a> but it is a multiple curve fit solution and it only gives significantly different answers at the extreme high and low end of star masses.


This is a great thing to calculate and store in a spreadsheet, which is what I do.