Moving either version should break the spell, at a DM's discretion
As Jack points out, Glyph of Warding has nearly the same language. The designer intent of this language (for Glyph of Warding) as shared by Mike Mearls in a tweet was:
if you can move it, it's an object can can't [sic] go more than 10 feet from casting point
Presumably this means; "If you can move the thing it is cast on, then it is an object and thus cannot go more than 10 feet from the casting point"
As Thomas inferred, in practical terms the surface is the option that cannot be moved, so you need the movement clause for only the object case.
However, as a table would normally qualify as an object under game term definitions, this caused problems and confusion.
Glyph is much more heavily played1 and thus has been a source of these questions, while Symbol has not. Its language has been errata'ed in 2016 to include the following sentence:
If the surface or object is moved more than 10 feet from where you cast this spell, the glyph is broken, and the spell ends without being triggered
The errata solved the issue by making sure movement breaks the spell regardless of what it is cast on. So why did they not apply the same errata to Symbol?
One could argue that Symbol is much higher level, and therefore moving it is an option that was consciously left in. Because frankly, you can upcast Glyph to any level and cause effects with it that are just as nasty as Symbol's, so what is the point of Symbol at 5 times the gp cost, to conserve an extra slot when prepping the effect?
Or, leaving out the movement clause could simply have been an oversight, as Symbol did not come up. We cannot know for sure.
Strictly as written, you could move it if you chose the surface option. That makes little logical sense, as you point out yourself. So, you as the DM need to decide how you want this to play out.
Personally, I think the lack of errata was just an oversight, and you should play it as if it would also break on surfaces when moved, but you can decide against it and follow the text.
P.S. If you really wanted them to bring pre-cast symbols along, they could cast them on the surface walls of a portable hole or Demiplane. I think these are pretty "gamey" tactics, though.
1 90% of play happens in tier 1 and 2, where players can use Glyph but not yet Symbol.