yoga asana - the moving aspect - is only 1 of 8 branches of yoga and it has evolved significantly over time.
Even BKS Iyengar, the man who brought yoga to the West, changed his practice over time from a strong, powerful and probably sweaty style when he was younger to a slower style as he aged.
there isn't a "right way" to do yoga, but if you're looking for authenticity then follow a lineage that has ties to India. Iyengar yoga is probably the closest in the US and even those expert teachers will have different styles.
at any rate, find an instructor that gives detailed instructions. you shouldn't have to look at your neighbor to figure out what your body ought to be doing, and the point isn't for your hand to reach that one hard-to-reach place on day 1. that's why there are props because everyone's body is different.