I think you are asking about the difference between risk types relative to the CAPM.
1 there are two types of risk, unsystematic risk or rather risk that can be diversified, and systematic risk which is basically the risk of change in market conditions causing return fluctuations.
2 the total variance or rather risk is a combination of the two
3 CAPM only calculates the expected return of an asset based on systematic risk which is measured by beta.
“it is the portion of asset j’s risk, contributing to variation in the market portfolio’s return.”
This sentence is missing details but based on my reading of it, the book is likely saying that the risk held by asset j causes an increase in total risk of the market portfolio, and does not help to diversify away its unsystematic risk.
Usually, when an asset is added to a portfolio, and that portfolio is quite large, it is possible that the unsystematic risk is diversified away, leaving only systematic risk. This is because, company risk of one asset can be countered by the risk of another asset when they have negatively correlated returns.
“ While the β of asset j is more typically referred to as asset j’s systematic risk measure it more precisely measures the systematic risk of asset j relative to the systematic risk of the overall market.”
As for this statement, it is stating that beta of assets measures the tendency for when there are movements of the market, how the asset is likely to move or rather, how much is this asset affected by a change in market conditions compared to most stocks in that market.
In short, the difference between definitions in your book, from my reading, it that of the types of risk. Adding a stock with high systematic risk to a portfolio increases its variation levels. Similarly, if that asset is positively correlated with the other stocks and has high variation in returns, not due to market conditions, that asset contributes to a portfolio’s total risk/ variation.