#Mathematica 138 bytes

Not finished! This works for all cases except [[(0,0),(1,2)],[(0,0),(2,1)]]

Length@Union[Join@@(Cases[RegionIntersection@@# &/@Subsets[Line[{{#,#2},{#3,#2},{#3,#4},{#,#4},{#,#2}}]&@@@Flatten/@#,{2}],Point@a__:> a])]

Example

Length@Union[
Join @@ (Cases[RegionIntersection @@ # & /@ Subsets[
Line[{{#, #2}, {#3, #2}, {#3, #4}, {#, #4}, {#, #2}}] & @@@ Flatten /@ #, {2}], 
Point@a__ :> a])] &@{{{-8, 6}, {-4, -2}}, {{-4, 9}, {4, 3}}, {{2, 10}, {14, 4}}, 
{{1, 7}, {10, -6}}, {{7, 4}, {10, 2}}, {{5, 2}, {9, -4}}, {{-6, -4}, {-2, -6}}}

6

Mathematica 138 bytes

Not finished! This works for all cases except [[(0,0),(1,2)],[(0,0),(2,1)]]

Length@Union[Join@@(Cases[RegionIntersection@@# &/@Subsets[Line[{{#,#2},{#3,#2},{#3,#4},{#,#4},{#,#2}}]&@@@Flatten/@#,{2}],Point@a__:> a])]

Example

Length@Union[
Join @@ (Cases[RegionIntersection @@ # & /@ Subsets[
Line[{{#, #2}, {#3, #2}, {#3, #4}, {#, #4}, {#, #2}}] & @@@ Flatten /@ #, {2}], 
Point@a__ :> a])] &@{{{-8, 6}, {-4, -2}}, {{-4, 9}, {4, 3}}, {{2, 10}, {14, 4}}, 
{{1, 7}, {10, -6}}, {{7, 4}, {10, 2}}, {{5, 2}, {9, -4}}, {{-6, -4}, {-2, -6}}}

6