I am using fuel oil to transport timber from my own forest for my own use and carry out transport related to a sale at delivered price from my own farm.

You can transport timber from your own forest for your own use or carry out transport related to a sale at delivered price from your own farm without paying a daily fee, if you are a person engaged in agricultural or forestry work.
You cannot transport timber that is a part of a sale at delivered price from another farm with a tractor you own. The tractor must be jointly held by a person or persons engaged in agricultural or forestry work. Who actually drives the tractor does not matter.

My child uses a tractor/tractor all-terrain vehicle for trips to and from school as well as going for a drive.

A tractor/tractor all-terrain vehicle can be fuelled with fuel oil for trips to and from school as well as going for a drive by paying the tractor’s daily fee.

I have timber sawn for my own use and transport it with my tractor to the farm I own.

A person engaged in agricultural or forestry work can use fuel oil to transport timber to the operator’s own farm without paying a daily fee, if the timber is intended for the use of the person’s private household.

I transport timber from the sawmill based on an assignment by the sawmill.

The daily fee of a tractor must be paid for a transport carried out with fuel oil if the tractor is used to transport sawn timber from the sawmill to buyers based on an assignment by the sawmill; in that case, the transport cannot be considered driving related to agriculture or forestry, but instead a transport related to the industrial processing and sale of timber.

I am a part-time farmer and receive my main earned income from another profession, and I transport my own timber for my own use.

If you are a person engaged in agricultural or forestry work, you are allowed to use fuel oil in transports related to your private household without paying a daily fee. For example, if your single-family home is not located in connection with the farm, you can transport firewood from the farm or a forest plot to your single-family home by using fuel oil.

I am transporting an excavator to the construction site of an attached house on a tractor carriage.

A daily fee must be paid for transporting an excavator to the construction site of an attached house by using fuel oil, because this does not constitute agricultural or forestry transport. However, you are allowed to transport an excavator to the farm by using fuel oil without paying the daily fee, if the work is done at the farm.