Simple question: What do you have set in the MCP when you first get airborne? I suspect it is 4000ft, which you will amend to 5000ft and then 6000ft as you pass the various DME points. Or maybe VNAV will do it for you.
Whatever, you are initially climbing to 4000ft and that is what you should report. The reasons for it have already been given.
By the way, you are not "cleared to 6000' on the SID ", bekolblockage. The SID chart explicitly shows mandatory level-offs at 4000ft and 5000 ft, unless (verbally) cleared by ATC.
Your argument would be like saying that just because a STAR shows an expect level of FL80 by HON into EGBB that you should call that as your cleared level even if you are actually still at FL380.