Howdy and welcome to the forum!
I will warn you in advance that I'm going to run off on a sort of educational tangent that may be confusing to you at this point. For the simple
answer, skip to the bottom, in blue
Really, it depends greatly on how the horse you're riding was trained. Some folks do train the way you are describing. HOWEVER, many folks don't
and the worst part is that it can get so complicated depending on the type of turn you want; rollback, forehand turn, or turning around an obstacle like a barrel or a tree.
Most people ride the way that I do, where you want a horse to move away from pressure when it's applied anywhere on their body, therefore, they should move away
from whatever leg you push them with.
It also depends if you're talking about neck reining or direct reining. Judging from how you described it, I suspect you aren't
neck reining your horse.
I'll try to describe in detail how I turn a horse for each type of turn to the right
and using direct rein like you are, so in order to get a left turn, you just reverse everything..
Rollback, where they keep their haunches set in one spot and swing their front end around to face the other way: I use a little bit of right rein and apply a firm left leg to move their shoulders over.
Forehand turn, where they keep their front feet planted and spin their hindquarters around until they're facing the other way: I don't use rein for this except to keep them from moving forward. To move their hindquarters only around to the left (which would actually turn their head to the right, turning their body clockwise), I apply right leg a little closer to where the back cinch lays so that they know I'm asking for the hindquarters and not the ribs.
Bent turn to go around an obstacle like a barrel or a tree: I'll take up a little bit of right rein to bend their neck to the right, I'll apply right leg up next to the girth to get them to bend their ribcage outward to the left, and I'll apply left leg back toward the back cinch to keep them from swinging their hindquarters out.
That will put their body shape similar to this and doing this will keep them from banging your inside knee on whatever you're going around.
The reason that it can get so complicated is that on a finished horse, there are usually at least 3 positions on their side; each controls a different part of their bodies. The most frontward position controls their shoulders, the middle position controls their ribcage, and the position farthest back controls their hindquarters.
BUT, all that's really beside the point and delving deeper into the art of riding a more finished horse. Probably more information than you really needed and is leaving you feeling really overwhelmed and for that I'm sorry. Put simply, most of the riders I know use their legs in the opposite way than what your grandpa described. Most riders, to turn right, will use right direct rein and left leg or, to turn left, will use left direct rein and right leg.