How do I negotiate a better...job title?

Another company made a job offer and I’m preparing to sign it and switch over to them.

After much negotiating, they have agreed to offer me the same salary. The only issue is that they’ve offered me a job title that I’ve already moved past – they want me to come in at a “coordinator”-level title.

I’ve only been in the professional world for 2 years out of college, but I have started at my previous company as a “coordinator” and was later promoted to a more mid-level position.

I feel like going back to a “coordinator” would be taking a step back. Am I wrong to assume this? Is there a tactful way to negotiate my job title without sounding vain?