The total risk of an asset can be decomposed into systematic (non diversifiable) risk and non-systematic (diversifiable) risk.
diversifiable risk is the component of total risk that can be be eliminated through portfolio diversification or holding a collection of assets which are less than perfectly positively correlated (less than +1 correlation) with each other. Since the investor can eliminate this risk component, he will not be rewarded for taking on this risk when making investments.
non diversifiable risk is the component of total risk that affects all assets in one way or another (war,inflation,politics) hence you cannot eliminate it through diversification. The market will reward the investor for assuming systematic risk.
The CAPM beta is a measure of systemtic risk with respect to market movements in the sense that general market movements affect all assets in one way or another, generating a risk that cannot be removed through diversification, and hence demands compensation that scales with exposure.
Hence we have CAPM : rf + beta*[E(Rm)-rf]
beta*[E(Rm)-rf] is the risk premium that the market provides to the investor for assuming systematic risk. The more sensitive the asset return is to general market movements,the greater is beta,the greater the compensation or risk premium.
If beta were a measure of non-systematic risk,as you thought, then it would not appear in the CAPM because its effects on portfolio risk can be diversified away.
The beta of a portfolio is simply the weighted average of all betas associated with the assets in that portfolio. So you can reduce portfolio beta by combining asset1 with beta=2 and asset2 with beta= -1 .This is somewhat different from the diversification argument.
When you want to reduce portfolio beta, you want to find assets that when combined into a portfolio would reduce the systematic risk (average portfolio Beta).
When you want to diversify a portfolio, you want to find assets that when combined into a portfolio would eliminate non systematic risk.
In the former case you are reducing portfolio beta. In the latter case you are reducing portfolio risk by eliminating non-systematic risk. Hence reducing systematic risk (portfolio beta) by combining various assets into a portfolio should NOT be confused with diversification.
Hope this helps.