The Execution Report message is used as a response message to the client, which contains information about a particular order in some certain scenarios such as:
confirm the receipt of an order
confirm changes to an existing order (i.e. accept cancel and replace requests)
relay order status information
relay fill information on working orders
relay fill information on tradeable or restricted tradeable quotes
reject orders
report post-trade fees calculations associated with a trade
If you are to represent an order which is submitted from a FIX Client to a FIX Server, the best approach to use the New Order Single message that you have mentioned
FIX message types are asset class agnostic, there are no special message types for the entry of an equities order. Thank you to @harinda05 for explaining the ER message.