I was curious myself, so I've ran a quick benchmark:

  import random
  import timeit

  heights = [random.randint(0, 10000)/100 for i in range(10000)]

  def benchmark1(heights):
      smallest = heights[0]
      count = len(heights) - 1
      while count > 0:
          if heights[count] < smallest:
              smallest = heights[count]
          count -= 1
      return smallest


  def benchmark1b(heights):
      a = 1
      b = len(heights) - 1
      min_height = heights[0]
      while a < b:
          if heights[a] < min_height:
              min_height = heights[a]
          a += 1

      return min_height


  def benchmark2(heights):
      smallest = heights[0]
      count = len(heights) - 1
      while count > 0:
          smallest = min(heights[count], smallest)
          count -= 1
      return smallest

  print(timeit.timeit('min(heights)', number=1000, globals={'heights': heights}))
  print(timeit.timeit('benchmark1(heights)', number=1000, globals={'heights': heights, 'benchmark1': benchmark1}))
  print(timeit.timeit('benchmark1b(heights)', number=1000, globals={'heights': heights, 'benchmark1b': benchmark1b}))
  print(timeit.timeit('benchmark2(heights)', number=1000, globals={'heights': heights, 'benchmark2': benchmark2}))

Here are the results in Python 3.11:

  0.04471710091456771
  0.21777329698670655
  0.22779683792032301
  0.6679719020612538

So, using min over a list is ~5x faster, using a single variable and a constant 0 is ~5% faster than using two for boundaries, and using min inside the loop instead of the if check is another 3 times slower: so, the old approach of looking for opportunities to use a builtin instead of looping still likely "wins" in the newer interpreters too, but if someone's got 3.14 alpha up, I'd love to see the results.

I might install 3.13 to check it out there too.