The exact answer to your question is found in a 500 page document titled "United States Standard for Terminal Instrument Procedures (TERPS)" that was revised and issued as Order 8260.3C on 03/14/2016. This is the answer to the part of your question about "where does the FAA define this."
As mentioned, the FAA uses templates. There used to be 31 templates of various sizes that are identified by pattern numbers 1 through 31. However, Patterns 1-3 are no longer in use. So there are now only 28 in use. Originally, these templates were literally plastic transparent templates. Now, most holding patterns are constructed using automation (i.e. computers). The following figure shows the exact proportions and geometry of every one of the 28 holding patterns in use today.
Point "L" (little orange circle) is the holding fix. This is a standard right hand holding pattern, (left hand holding patterns were drawn by flipping the plastic pattern over for a symmetrical left-handed pattern). The solid black line that runs from point "M" to point "L" is the holding course. All the straight line segments are defined by the definition of the respective Pattern Number as provided in the following chart.
Take a look at the dimensions of Pattern Number 4 on the chart above. Pattern 4 is the smallest holding pattern in use today. Specifically, see that the column M-E says 5.3 nautical Miles, and that column L-I/M-H reads 3.5 nautical Miles. Now look at the template in the figure above and find the line M-E. This is the protected distance from the holding course on the Holding Side of the pattern. Now, find the lines L-I and M-H. These represent the protected distance from the holding course on the Non-Holding side of the pattern. This is the answer to the part of your question about "what are the dimensions of the holding patterns and how does the FAA define it."
Now my rant. I often hear pilots and even flight instructors talk about the "Protected Side" and the "Non-Protected Side" of the holding pattern. This is inaccurate. The FAA does not describe holding patterns in that way. There is either a Holding Side or a Non-Holding Side. BOTH sides are protected! Pattern #4, the smallest pattern in use in the USA, has 3.5 nautical miles of protection on the Non-Holding side. I plugged in all of the values in the chart above into Excel and calculated the ratio of Non-Holding Side protected area to Holding Side protected area, and all of the patterns had a protected area on the Non-Holding Side equal to approximately 2/3 or 67% of the Holding Side. That may not be as much protection as on the holding side, but it translates into multiple miles of protection.
Next time someone says Non-Protected Side of a holding pattern, take the time to explain to them that both sides are protected. Protection on the Non-Holding Side is necessary to protect aircraft entering a holding pattern using a parallel entry with a cross-wind that is pushing them away from the holding course into the Non-Holding side during that initial 1 minute outbound entry.
End of RANT