First of all, welcome to the forums! I hope they'll be a helpful resource to you as you begin your journey into the world of magic.
Yes, there are all kinds of ways that this effect can be done, though most would use the same basic principle. It's pretty much the most basic beginner's card trick. Which isn't a bad thing, it's a good starting place, and presented right, it can make a great trick.
I'd also like to note -- since you're not counting down the number of cards, why do you have the spectator return the card that way? That's the explanation that will immediately jump to their mind. It doesn't matter if they're wrong, it still isn't magical to them. This is true even when their "easy" explanation wasn't even possible, but in your case it's very clearly possible, and would actually be pretty easy for just about anyone to do. Better to have them cut the deck themselves, place their card in the middle, and complete the cut. Then maybe do a few more cuts so no one really knows where it is. That will seem a lot more fair to a lay person.
Rather than trying to come up with your own effects right now, I'd suggest you pick up Royal Road to Card Magic and study that. There's a lot of good material in there right off the bat that will soon have you performing cleaner, more convincing versions of the effect you're trying to achieve here.
On another note, lay people don't really care how they pick their card. If you do something strange to have them select it, they'll assume that you're trying to cover something sneaky, even if you aren't. If you want to make it obvious that it was a completely free choice, I'd just hand them the deck and have them remove a card themselves. Through the methods you'll learn in Royal Road, you'll be able to have them return the card to the deck in a much more natural manner to set up this effect, in addition to learning the fundamentals in other areas of card handling.
Finally, the effect of turning over cards face up until you stop on their card just isn't that interesting, to be honest. Royal Road, Mark Wilson's Complete Course in Magic, Card College, Expert at the Card Table -- these books will help you learn effects that have more engaging, fun ways of revealing the chosen card. Some of these you'll be able to start performing right away, others will require some practice.
If you are going to perform this type of effect, one way you can make things more interesting is throwing in a simple "magician in trouble" plot. When you see their card, don't stop. Count past it a few cards, and then slow down, and say confidentially that the next card you turn over will be their card. They'll assume you mean the next card in the deck, and depending on the person, they'll either feel bad for you, or they'll think this is hilarious. You then reach to the pile of face up cards you've dealt, pull out their card, which you noted earlier, and turn it face down onto the table. A lot of people will have seen this version of the trick as well, but it's still a more engaging presentation.
