Read a little further:According to Wikipedia: "Rent-seeking means seeking to increase one's share of existing wealth without creating new wealth".
Rent-seeking is an attempt to obtain economic rent (i.e., the portion of income paid to a factor of production in excess of what is needed to keep it employed in its current use) by manipulating the social or political environment in which economic activities occur, rather than by creating new wealth.
That is a key point.
Increasing the rent on a property I own increases my wealth without creating new wealth.
You are confusing rent with, well, rent. Increasing the rent on your property gives the tenant the opportunity to decide wether the value they receive from the property is worth the cost or they should move elsewhere. If you are confident the property can be rented at the higher rate then that is the price that needs to be paid to keep it in production (as pointed out in your clip from Wiki); and thus is not rent seeking even if you increase the rent.
Your earlier example of property value increase above inflation is not rent seeking either, as the return needs to be risk adjusted in order to be a viable investment.