新加坡过境免签的困扰？？？急问大虾们！！！

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我最近准备回国一趟，想过境新加坡找一朋友玩，请各位大虾评评我的情况能不能拿到新加坡的96小时转机免签入境许可？？

我持有1年的有效的美国入境签证（B1/B2）并且有三年的加拿大工作许可(Work Permit)。在网上找到一个便宜的回国机票路线：
纽约—北京（转机）—新加坡（转机过境3天）—成都—香港（转机过境2天）—纽约

我的问题是，我在北京转机还算不算从*第三国*免签入境新加坡？？我需要在北京入境中国吗？？（我不想在北京入境）如果必须在北京入境，没有新加坡签证北京还会不会放行？？

以下是从新加坡旅游局网站摘录的有关过境免签的文字：
http://www.yoursingapore.com/content/traveller/en/plan-your-trip/travel-essentials/visa-customs-legal-regulation/visa-free-transit-facilities.html


PRC and Indian nationals who are in transit to or from any third country by air will be eligible for the 96-hour visa free transit facility (VFTF) if they are in possession of a valid onward air ticket departing within the next 96-hours and satisfy the following criteria:

Possess a valid visa/long term pass (validity of at least 1 month) issued by any of the following countries:

a.Australia
b.Canada
c.Japan
d.New Zealand
e.United Kingdom
f.United States of America
g.Germany
h.Switzerland

请各位赐教！