Those aren't shoes that'll take an SPD aka a two bolt cleat.
What you have linked are for three bolt cleats aka a road cleat. They are likely good for spin as they have a solid sole that doesn't flex, that's better for power transfer. If you have a flexible sole some of the power from your legs will be wasted into bending the shoe. However, I can't really see how that's important in a spin class. If you are racing (indoors or not) the power transfer and not wasting it is important!
You might wanna check first what the peddles at your gym will actually accommodate. Then take it from there!
I'd be inclined to go for more of a mountain bike shoe (if the peddles will take an SPD/ 2 bolt cleat). To be honest the recessed cleat is a bit of a red herring... you'll still go slip sliding! I'm speaking from bitter experience as a mountain biker (taking cafe stops), cycle commuter and cycle tourer.
The other issue you'll have is setting the cleat up correctly on the shoe. That will be a problem with both 2 and 3 bolt. If it isn't just so, you'll end up opening the door up for injuries as your legs won't be tracking round correctly. It's fairly simply overcome - just buy yourself the correct ergon cleat tool. You need to be aware this is an issue though and it's important the cleats are set correctly.
You may just be better off sticking with trainers and toe clips. Cheaper. Less margin for error!