ie. the issue is not with how we define the second. The issue is expecting that if two people in different frames of reference wait for one second, that they will continue to be "synchronised".
In fact it's worse than that, it's impossible to even define the concept of "synchronisation" for entities in different locations. You can only know that two things are synchronised if they are in the same location.
If the two entities are in the same frame of reference (ie. not moving or accelerating relative to each other) then you could argue that you just send a beam of light there and back, and assume the light takes the same time in each direction in order to synchronise... But we don't even know that light travels at the same speed in all directions! We've never been able to measure the "one-directional" speed of light, we can only measure the round-trip, so for all we know light could have a preferred direction in the universe and travel faster in that direction, as long as it is the same amount slower in the other direction. We would never be able to tell.