The halving will take its effect in the next few months or years. The price will rise due to the reduced supply.
The supply is not reduced. It increases toward 21 million with every block.
The daily supply of the new coins are reduced, or you can say the rate of the new bitcoin growth reduces.
I see that you have a full understanding, but "daily supply" is a poor term because the supply of bitcoins each day includes all bitcoins in existence, and not just the ones produced that day.
"Production" is a better term than "supply of the new coins" because it is more accurate and accepted, and it isn't as confusing.
It could go both ways, as the coins mined are the only 'supply' to the miners trying to make money. Really it just comes down to semantics.
Anyway, I've been saying that nothing much was going to happen to the price for a while. Everybody knew when the halving was going to happen and bought coins accordingly, so there really was nobody to make the price go up when it actually happened. I could see it potentially being something that helps the price slowly grow to a point that the bull market takes over on it's own, but not something that's going to shoot us into a new bubble without any hype.
We're also going to need to overcome all the people that overbought BTC in preparation for it and are waiting for a good point to cash out. The second we hit a landmark amount like 700, there's going to be a lot of sells from people just trying to make a quick buck. Only after we overcome that can we actually discuss a huge price raise.