the MD-PSTP linked above is an umbrella program encompassing the various physician scientist pathways. I dont think its a distinct entity you apply to or anything like that. If it is, I've never heard of it being mentioned distinctly from the programs that make it up.
There are many ways to do a "5 year program" here. The one mentioned on the website is splitting the curriculum where you essentially do MS2 at "half-speed", freeing up 50% of your week in your new extended MS2 to do research, so 5 years total and 2 years of research at 50% effort, nominally (and extends your preclinical phase to 3 years). You apply to split after first year and propose a project / reason for doing so (like research or getting an extra degree). You could also do a more traditional research year between MS2 and 3 or between MS3 and 4.
Berg is a 6 yr path with funding from BWF, funded like the MSTP starting after year 3. Berg students are required to split (see above) as it currently stands.
The MSTP is the MD/PhD program, more or less self-explanatory.