At the user level, typically 24 mg/ml and below, people can expect occasional contact with eliquid, it's a part of life. If it's a small quantity, most people are absolutely fine just wiping it off with a kleenex or paper towel; we are, after all, nicotine addicted, have at least some level of tolerance to it. (people who regularly vape 36 mg/ml likewise have a tolerance for higher strength, the same kleenex kind of approach likely is fine)
Larger spills, or higher mg (higher than you normally use), absolutely go for the washing.
Working with 50 - 100 mg you should be a DIY person, which means read the precautions, follow appropriate safety procedures.
I just had a full 6ml daily-use bottle go through a full wash/dry cycle, unintentionally, at home. It was absolutely fine. Guess I'll stay with the bottles I use
The abstract of the article referenced above doesn't state that washing it off is bad, but rather "The length of contact time affected the amount of substance passing the skin, ... This emphasizes the importance of washing away nicotine spilled on skin rapidly. " In other words, if you get a bunch of nicotine on you, by all means wipe/wash it off. It's important to get this stuff straight.
I always have been, and remain, more concerned about keeping eliquids away from kids and pets than the little bits I get on me from vaping. Others don't have nicotine tolerance. This includes making sure that any mop-up tissues/kleenex aren't in places accessible to others.