I have converted the transfer function to the standard format but I couldn't fully write the equation for the phase. please check whether the Bode plots are correct.
There has come to be two different meanings of what a "Bode Plot" is.
The original Bode Plot was an asymptotic approximation to the frequency response the creation of which was probably the outcome of the era where computers were not known and so approximations where considered a calculatory necessity and were accepted with open arms. There were small adjustments made to the original Bode Plot to account for higher order effects, but the majority was based on straight line approximations. After the advent of computers the original idea of the Bode Plot faded a little because now we were able to plot the actual frequency response directly from the transfer function without going through an approximation scheme. This of course allowed more accurate plots to be done, but took away the insight that can be gleaned from knowing how the asymptotic approximations work.
This means that the plot in post #1 has been done in the form of the original Bode Plot, while the plot in post #2 is just a frequency response plot even though today #2 may be referred to as a Bode Plot also.
In the asymptotic approximation or straight line approximation as we call it sometimes the behavior of the frequency response is known to follow some very basic and simple rules at the critical frequencies, and following those rules we can often see the picture of the frequency response without actually plotting it. That's the beauty of the method. In the pure frequency response plot we must calculate each and every point of interest, but that produces a very accurate picture of the response.
The main point though is we have to know which form we are talking about in order to comment about what is to be done and what is right or wrong. This leads to two different approaches.
For the straight line approach, we look up the rules for the critical frequencies, then apply those rules.
For the exact frequency response plot, we convert the variable 's' nito "j*w" and then often 'w' to "2*pi*freq" and then simplify the equation, then find the amplitude and phase shift, then plot both using some software.
Because the two methods produce similar plots, we can use one to check the other.
I'll try to get to check the waves a little later today unless someone else gets to do this first.