By “truth,” I don’t wish to beat a dead horse in terms of the whole “objective vs. subjective truth” line of discussion.
Rather, I mean can there be different versions of objective truths? (Or facts)
A larger "horse" will have to be beat a little bit in terms of what "truth" is and is not, to understand why it is "yes".
"Truth" is not a synonym for existential affairs and happenings which are not a representation. "Objective truth" is not a synonym for existence that is not a representation.
So it's unavoidable that different domains of human endeavor (systems, practices, cultures, etc) are going to output representations of human experiences, sensations, and inferences about them which vary (they will not be like identical products cranked-out by an assembly line at a particular factory).
Representation = cognitive orientation, description, account, portrayal, likeness, simulation, etc. Most if not all brain/body or human-outputted representations of things they perceive and think about are also synopses and generalizations. Vastly incomplete, since they can't include every specific detail.
A truth is a representation that is approved by whichever human agency (and/or direct environmental experience outcome) as accurate or satisfactory with regard to representing something deemed actual or that has been the case in the past (like Lincoln being assassinated). Truth may also refer to an effective principle that reliably predicts, manages, or brings about expected results if adhered to.
An absolute truth is immutable and universal. It's invulnerable to relationships (doesn't change via connection or contingent association with other things). In everyday encounters this "global and without exceptions" idealization will usually only apply to the territory of an institution, system, practice, etc that accepts and uses it.
Obviously "truths" with contingent, warranted, revisable, and other adjectives affixed to them belong to opposite or alternative species.
"Objective" and "subjective" are a couple of the many contexts or classifications that can apply to truths. The latter refers to a representation built around personal interests and viewpoints; or a group version of that if the members are in agreement but potentially in conflict with other groups. The former (objective) designates a representation without personal affairs and biases (or wherein there is an effort to at least minimalize such).