How can I accommodate newer washing machines with older drains?
Is there any significance to the second 2&quot; pipe at the outlet side of the trap?

No. The vertical pipe downstream of the trap is either another drain coming in from above or a vent stack. It isn't involved with the capability of the 3&quot; section to provide some surge capacity for the draining washer.
(Well, I suppose it does add a little bit of capacity, insofar as water can fill that drain/vent pipe up to the level of the opening at the top of the 3&quot; section. So that's a little bit of surge capacity. But it isn't its primary purpose.)