Tricolorability is a knot invariant, and the trefoil is knotted

Invariance under Type I. A Type I move adds or removes a small loop, creating one crossing where a single strand crosses itself. In any valid tricoloring before the move, the strand has one color, say $a$, entering and leaving the loop. Adding the twist creates a crossing with under-strands both colored $a$ and over-strand also forced to color $a$ (since the loop is a single strand), and $a+a+a=3a\equiv 0\pmod 3$ always holds automatically — so the coloring extends validly to the new crossing without constraint, and conversely restricts validly when removing it. The overall colorability (existence of a valid coloring using $\ge 2$ colors) is preserved.

Invariance under Type II. A Type II move introduces or removes two crossings where two strands, colored $a$ and $b$ (possibly equal), cross twice. Checking both crossings: at each, the two strands present are $a,b$ and the third (the over-strand at each crossing, which is one of the same two strands continuing through) is forced by the rule to be whichever color makes $a+b+c\equiv0$; since the two crossings are related by the strands simply passing over each other, the coloring on the two original strands $a,b$ extends consistently to color both new crossing regions without introducing new colors or contradictions, and removing the move just deletes those two constraints, which were automatically satisfiable. So colorability is preserved both ways.

Invariance under Type III. A Type III move slides a strand across a crossing, rearranging three crossings among three strands colored $a,b,c$ (say) without changing which colors appear where globally — it only changes the local combinatorial arrangement of the same three colored arcs. Since the coloring rule $x+y+z\equiv0\pmod3$ at each of the three crossings depends only on the colors of the (unordered) triple of strands meeting there, and Type III does not change which strands meet at crossings (just their diagram-local arrangement), a valid coloring before the move restricts to a valid coloring after, and vice versa.

Conclusion. Since tricolorability (as a yes/no property, requiring $\ge 2$ colors used) is unchanged by all three moves, it is an invariant of the knot, not just the diagram. The standard trefoil diagram admits the coloring $a=0,b=1,c=2$ on its three arcs (one checks all three crossings satisfy $0+1+2=3\equiv0\pmod3$), using $3\ge2$ colors, so the trefoil is tricolorable. The standard unknot diagram (a single unknotted loop, no crossings, or any diagram reducible to one arc) has only one arc, hence only one color is available, failing the "$\ge 2$ colors" requirement — the unknot is not tricolorable. Since tricolorability is an invariant and the two diagrams disagree, no sequence of Reidemeister moves connects the trefoil diagram to the unknot diagram, so the trefoil is a genuinely knotted circle.