Not to rain on your parade, but the city health / sewer department may rain on it later with a forced 'FIX IT NOW' citation and/ or fine.

That outdoor 'drain' as you all it needs to be identified. If it is what your eaves trough dumps into you could be in for a rude awakenging. Your municipality might have a separate Rain water sewer and a separate waste sewerage system. Pumping untreated household waste, which is what you are discharging from your washer, needs to be treated along with your toilet flushings. Rain water run-off is usually directed to a nearby stream with little to no treatment.

The subject 'drain' next to your foundation may be dumping into your abandoned septic system only to cause problems later.

The subject 'drain' may be being plumbed into your foundation tiling and end up in your sump pump.

You need to determine where that pipe goes. In our little village, some older homes were plumbed directly into farmers' field tiles... it was the way they did it a hundred years ago, but quite illegal now.

I would suggest, if you are moving the plumbing anyway, plumb it into a waste line coming from an upstairs sink drain or toilet. That is where the sewerage belongs and will keep you out of hot water (pun not intended) with the municipal inspectors.

Just for reference, I have a basement washer / dryer. The was water dumps into a 1.5" vertical pipe that drops down to a P-trap, then under the concrete floor to a sludge pit. A sludge pit is similar to a sump pit, but has thick walls and a hermetically sealed lid. There are two holes in the lid. One for the pump to pump the effluent up and into the 2.0" drain, down stream from the kitchen sink which quickly dumps into a 4 or 6" drain out to the septic tank.
The second hole is plumbed into the vent stack to keep sewer gasses out of the basement. The pump is a 2" discharge sludge pump, capable of chewing up normal solids like TP and ... well normal toilet solids. (not feminine napkins) I've replaced it once in 15 years. Probably all it needed was a float switch, but I was young and ignorant back then. If it will handle TP, it should and does handle normal washing machine lint with out any complaints.

Sounds like you have it well in hand.

I remember helping a lady in town who thought she should have her septic tank pumped, but didn't know where it was. There is only septic in our village, no municipal sewerage. I and a friend did some inspection and determined that the cast iron waste pipe, which was in the front of the house. This is unusual as septic fields are at the back of the house. She had, maybe, 30 feet to the street from the front of the house. The waste line appeared to run directly across the street in a straight line to God knows were. The house was built a hundred years ago and at one time may have been the parsonage to the church across the street. We suspect, it might have been plumbed into a common septic tank / field with the church. So.. God does know.