How should I categorize bank transactions when the account purchases units of stocks or mutual funds on our behalf?

Is this a business or personal account?  Most investors have a brokerage account with a company, like at Fidelity or Schwab and details are tracked in that account so on your books you can just use asset accounts.

- Cash   

- Investments

When you add to your investment account, do this entry:

DR Investment account

CR Cash account

You could certainly have multiple investment accounts, if you wanted to keep separate on books also:

Account 1101 - Investment - Stocks

Account 1102 - Investment - Mutual Funds