Is there a change over the business cycle in how much of economic activity is market-based?
Yes, the two figures are different because that is how the economy works in a circular flow. To calculate the size of economy (GDP), we will find the total expenditure, total income or total production. In theory, all three calculations should converge at the same figure. However, in economies they may give different figures. It is a fault of whole concept of GDP that it does not take into account any transaction for which there is not a market price available. So, no matter how much gardening you do at your home, you are not adding up anything in the economy. So, whatever mutual understanding you have with you neighbor and whatever transactions you make, it won't be considered. 
In the second part of your example, you are spending 10 dollars and your employer is spending 50 dollars so that add up to 60 dollars. This happens because of circular flow, your income is somebody else's expenditure. 

Production for which there is no price will not be counted.