What does the position of the ON clause actually mean?

It determines the logical tables involved in the join.

With a simple example

SELECT w1.WidgetID,
       w2.SomeValue,
       wp.PropertyName
FROM   #widgets1 w1
       LEFT JOIN #widgets2 w2
         ON w2.WidgetID = w1.WidgetID
       JOIN #widgetProperties wp
         ON w2.WidgetID = wp.WidgetID
            AND wp.PropertyName = 'b'
ORDER  BY w1.WidgetID 

#widgets1 is left outer joined to #widgets2 - the result of that forms a virtual table that is inner joined to #widgetProperties. The predicate w2.WidgetID = wp.WidgetID will mean that any null extended rows from the initial outer join are filtered out, effectively making all the joins inner joins.

This differs from q2...

SELECT w1.WidgetID,
       w2.SomeValue,
       wp.PropertyName
FROM   #widgets1 w1
       LEFT JOIN #widgets2 w2 --no ON clause here
                 JOIN #widgetProperties wp
                   ON w2.WidgetID = wp.WidgetID
                      AND wp.PropertyName = 'b'
         ON w2.WidgetID = w1.WidgetID
ORDER  BY w1.WidgetID

#widgets2 is inner joined onto #widgetProperties. The virtual table resulting from that join is then the right hand table in the Left Outer Join on #widgets1

The same result can be achieved by using a derived table or Common Table Expression...

WITH VT2
     AS (SELECT w2.WidgetID,
                w2.SomeValue,
                wp.PropertyName
         FROM   #widgets2 w2 
                JOIN #widgetProperties wp
                  ON w2.WidgetID = wp.WidgetID
                     AND wp.PropertyName = 'b')
SELECT w1.WidgetID,
       VT2.SomeValue,
       VT2.PropertyName
FROM   #widgets1 w1
       LEFT JOIN VT2
         ON VT2.WidgetID = w1.WidgetID
ORDER  BY w1.WidgetID 

... Or alternatively you could re-order the virtual tables and use a RIGHT JOIN instead.

SELECT w1.WidgetID,
       w2.SomeValue,
       wp.PropertyName
FROM   #widgets2 w2
       INNER JOIN #widgetProperties wp
               ON w2.WidgetID = wp.WidgetID
                  AND wp.PropertyName = 'b'
       RIGHT JOIN #widgets1 w1
               ON w2.WidgetID = w1.WidgetID
ORDER  BY w1.WidgetID 

This is covered by Itzik Ben Gan here

... the JOIN conditions must follow a chiastic relationship to the table order. That is, if you specify tables T1, T2, T3, and T4 in that order and the JOIN conditions match T1 with T2, T2 with T3, and T3 with T4, you must specify the JOIN conditions in the order opposite to the table order, like this:

FROM   T1
       <join_type> T2 T2
                  <join_type> T3 T3
                             <join_type> T4
                               ON T4.key = T3.key
                    ON T3.key = T2.key
         ON T2.key = T1.key 

To look at this join technique in a different way, a given JOIN condition can refer only to the table names right above it or table names that earlier JOIN conditions already referred to and resolved.

but the article has a number of inaccuracies, see the follow up letter by Lubor Kollar as well.

If you look at the FROM clause syntax diagram you will see that there is only one place for the ON clause:

<joined_table> ::= 
{
    <table_source> <join_type> <table_source> ON <search_condition> 
    ...
}

What you find confusing is simple recursion, because <table_source> in <joined_table> above can be another <joined_table>:

[ FROM { <table_source> } [ ,...n ] ] 
<table_source> ::= 
{
    table_or_view_name ... 
    ...
    | <joined_table> 
    ...
}

To avoid confusion you should use parentheses in non-obvious cases (like your examples) to visually separate <table_sources>; they are not necessary for the query parser but useful for humans.