Is there a way to determine when I will have my 3rd attack with Guinsoo's?

Nope.

There is no visual indicator for when your next attack will trigger on-hit effects twice. The only visual effect provided is when Seething Strike (the phantom hit) activates. Below is a gif I took in practice tool to double-check this:

You also do not gain a stacking buff, apply a debuff, or trigger any other less-visible effects when stacking up to your third attack. The only way of knowing when you'll trigger your on-hit effects twice is by counting your auto attacks, as you have, or by simply watching for the visual effect Seething Strike performs when it activates.

The attack count that counts upwards to 3, at which point Seething Strike activates, only remains active while in combat. If you exit combat (by dying, or by not dealing or taking damage from external sources for 5 seconds) after attacking twice, then your attack count is set back to 0, and you must enter combat and stack your attacks back up to 3 to trigger a Seething Strike again.