Yes, you got the effects exactly right. The land value tax would have no direct effects on rent. That's why it's such an awesome tax: it does not distort economic activity. Unlike eg income tax which discourages working.

Pro-forma the land will still be in private hands, thus encouraging people to improve it. (Remember: buildings are not taxed, only the land.)

The government will be very interested in improving the value of land, eg by making saner infrastructure investments, because they automatically get more tax money to play with that way.