I would very much agree with your assessment of the situation. Once the washer has pumped all the water out of the tub, it's done pumping. Water left in that hose will run back down and back into the tub, leaving your clothes wet again.
I see three possible solutions:
A second run of the spin-cycle may get a reasonable amount of water out of the clothes and take less time in the dryer.
Find a floor drain. Run your washer hose to a 5-gallon bucket and affix the hose so it drains into the bucket. Cut a hole (3-4" should do it) in the bottom of the bucket near one edge to allow water out. Place the hole in the bucket over the floor drain. Weigh the bucket down with a cinder block placed across the top. Water will drain into the bucket then run directly into the floor drain. The bucket will provide a sump/cache to hold water flowing from the washer in case it won't drain quite as quickly as the washer pumps it out. The bucket will also provide splash containment to help prevent water from spreading too far across the floor. Potentially significant drawback: If the water comes out too fast and/or the bucket gets moved, you can end up with water spread all across the floor. This appears to be a rough, unfinished basement with a concrete floor. Some water on the floor isn't likely to damage the structure, you just have to ensure you're not storing anything on the floor that should be kept dry.
Put a check valve on the back of the washer between the washer's outlet and the existing hose. This will allow the washer's pump to force water out of the tub, up hill and into the drain. When the pump kicks off, the valve will seal the drain pipe preventing the water from flowing down hill and back into the tub.
Get the washer to drain below where the outlet from the washer is, this will allow the hose to drain into the sewer instead of back into the washer. It appears that one of these might be a clean out or simply be a capped off bit of drain plumbing with a trap:
If that's the case, you may be able to unscrew the lid and have the washer drain into one of them. NOTE: You MUST have a trap after the opened pipe. If you don't you'll get sewer gas backing up into the basement. Aside from being smelly, it's potentially explosive! If you're not sure, I'd recommend that you ask the landlord about the pipes.