Always use detergent oil in any internal combustion motor, only pumps can use non-detergent since there isn't any dirt produced by just the lubrication in a pump.
The older oils may not have had as good a detergent in their mix but all engines need the cleaning agents or detergents in their oil.
IT makes absolutely no difference to the mower, the oil pump or any other mechanical device weather the oil is detergent or non-detergent.
Forget all the rubbish you have seen on TV read in glossy magazines or heard on the radio.
It is all pig droppings, not even bull written by very cleaver people who have never done a day of science or engineering in their life.
The deciding factor on detergency is the type of filter. Detergents were added to oils to hold the impurities in solution as fine particles so that the paper oil filter can remove them easily without becoming clogged
End of story
Prior to the common use of disposable paper filters oil was either centrifugally filtered or the oil went into a deep tank to sludge out.
It is always a source of mirth that people who know absolutely nothing about the chemistry of oils will argue about detergentcy with violent passion.
What is important for mower oils are the anti corrosion additives, anti oxidation additives and water dispersal additives because the engine sits there for months not moving.
But I read it on Face book so it must be true cause he has 2,987,789 likes and 50,000 friends.
The annoying bit is if you look up detergent oil in wikki, the correct answers are all there
I only got 1 hour a week for 13 weeks to cover oils, lubricants , cutting fluids & coolants which was just enough to show we knew next to nothing about them
tractordata.com says...The 8-horsepower Kohler K181 replaced the K161 for the 1964 model year. The K181 was available throughout the production run of the 110.
Kohler K181S
gasoline
1-cylinder
air-cooled
305 cc [18.64ci ]
Bore/Stroke: 2.9375x2.75 inches [75 x 70 mm]
Power: 8 hp [6.0 kW]
Air cleaner: dry
Compression: 6.8:1
Rated RPM: 3600
Starter volts: 12
Oil capacity: 0.625 qts [0.6 L]
Any air cooled engine oil will be fine and any mower oil will be even better and better still if it gets drained at the end of every season so over winter when corrosion happens you have nice clean oil in there.
Viscosity according to the local conditions.