I don't see how either of the articles you point to say that the claims made in the Washington Times article are debunked - they support what was claimed. The second article you cite explains very clearly that the City Attorney stated that if the pastor refused to perform a same-sex marriage that he would be in violation of the ordinance - violation of the ordinance is a criminal misdemeanor with up to 6 months of jail time for each offense and up to a $1,000.00 fine for each offense. That someone wants to earn a living through the performance of a religious act does not strip the act of its religious nature.
Catholics, for instance, view marriage as a sacrament and The Catechism of the Catholic Church defines the sacraments as "efficacious signs of grace, instituted by Christ and entrusted to the Church, by which divine life is dispensed to us. The visible rites by which the sacraments are celebrated signify and make present the graces proper to each sacrament. They bear fruit in those who receive them with the required dispositions."
The apostle Paul wrote about earning a living through the spreading the gospel:
"Who serves as a soldier at his own expense? Who plants a vineyard and does not eat its grapes? Who tends a flock and does not drink the milk? . . . Don't you know that those who serve in the temple get their food from the temple, and that those who serve at the altar share in what is offered on the altar? In the same way, the Lord has commanded that those who preach the gospel should receive their living from the gospel. " (1 Corinthians 9:7,13-14 - for the larger context see 9:1-14)