The important point is where you will pass through immigration.
The 144-hour visa-free transit requires that you stay within a certain area of your arrival point, and depart the country from within that same area. For Beijing, that means that you must stay within the Beijing/Tianjin/Hebei area - which obviously does not include the vast majority of the distance to Hong Kong!
As you've said, there are a few options for trains to Hong Kong. The high-speed "G" trains which take around 9 hours, and the conventional "Z" trains which take around 24 hours.
Other than the speed, these trains are also different in where you pass through immigration. For the high-speed G trains, both Chinese and Hong Kong immigration occurs in Hong Kong. This means that these trains are NOT eligible for the 144 hour visa-free transit, as you are travelling across much of China before officially exiting the country - which is not allowed under this visa-free transit.
However for the slower Z trains, you pass through immigration in Beijing before departing - which meets the requirements of the visa-free transit. Thus it IS possible to catch these trains to Hong Kong using the 144-hour visa free program.
The 144 hours visa program is only available to citizens of 53 countries, which does not include Turkey. However, as long as you are a citizen of one of those countries then having a Turkish visa in your passport should not be an issue. I have multiple Turkish visas in my passport, and have never had a problem entering China.