[Python] How do determine inner border points of polygon intersection

I am trying to write a Python script that will return me all vertice locations (x,y) of inner intersecting edge between a square tile and a closed polygon with n-vertices. I got this far with my code, but am getting wrong inner border vertices. Is there an algorithm of somekind specificaly to determine inside edges, so I know in which direction to search for.

from shapely.geometry import Polygon, LineString, MultiLineString
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon as MplPolygon

square_coords = [
(53906250, 35456250), # lt
(54296875, 35456250), # rt
(54296875, 35546875), # rb
(53906250, 35546785) # lb
]

polygon_coords = [
(54340742, 35518900),
(53997222, 35518800),
(53997298, 35183478),
(54340666, 35183478),
(54340742, 35518890)
]

square = Polygon(square_coords)
polygon = Polygon(polygon_coords)

intersection = square.intersection(polygon)

fig, ax = plt.subplots()
ax.set_aspect('equal')

square_patch = MplPolygon(list(square.exterior.coords), color='blue', alpha=0.4, label='Square')
ax.add_patch(square_patch)

polygon_patch = MplPolygon(list(polygon.exterior.coords), color='green', alpha=0.4, label='Polygon')
ax.add_patch(polygon_patch)

if not intersection.is_empty:
if intersection.geom_type == 'Polygon':
inter_coords = list(intersection.exterior.coords)
inter_patch = MplPolygon(inter_coords, color='red', alpha=0.6, label='Intersection')
ax.add_patch(inter_patch)
elif intersection.geom_type == 'MultiPolygon':
for part in intersection.geoms:
inter_coords = list(part.exterior.coords)
inter_patch = MplPolygon(inter_coords, color='red', alpha=0.6, label='Intersection')
ax.add_patch(inter_patch)

polygon_boundary = polygon.boundary
intersection_boundary = intersection.boundary

inner_border = polygon_boundary.intersection(intersection_boundary)

if not inner_border.is_empty:
if isinstance(inner_border, (LineString, MultiLineString)):
if isinstance(inner_border, LineString):
lines = [inner_border]
else:
lines = list(inner_border.geoms)

for line in lines:
x, y = line.xy
ax.plot(x, y, color='black', linewidth=2, label='Inner Border')
ax.scatter([x[0], x[-1]], [y[0], y[-1]], color='purple', s=40, zorder=10)
ax.text(x[0], y[0], f"({x[0]:.5f}, {y[0]:.5f})", fontsize=7, color='purple')
ax.text(x[-1], y[-1], f"({x[-1]:.5f}, {y[-1]:.5f})", fontsize=7, color='purple')

ax.set_title("Polygon-Square Intersection & Inner Border")
ax.set_xlabel("X")
ax.set_ylabel("Y")
ax.grid(True)
ax.legend()
plt.tight_layout()
plt.show()


Two examples of what I am trying to achieve. My goal is to get all intersecting vertices (x,y) on yellow border.





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