I understand you can take money out of a Roth IRA after 5 years to use for the purchase of a 1st house without paying taxes/penalties, but I'm not sure when that 5 year period begins. IRS Publication 590 defines it as "after the 5-year period beginning with the first taxable year for which a contribution was made". However, since you are able to make a contribution up until April 15 and still report that contribution on the previous year's tax return, I need some clarification.

I opened my Roth IRA around March 2004, so I was able to report those contributions on my 2003 tax return. According to Publication 590's definition, does this mean my 5 years starts on January 1, 2003 (the "1st taxable year")? When exactly can I withdraw funds and be past the 5 year point?

You are correct that your five year period started on 1/1/2003 and you will be past the five years at the beginning of 2008.