Best Way to Log Yard Work?

I don't think there is a way to figure that accurately since there are too many variables. For example, are you mowing a flat yard or a hill, are you using a self propelled mower, how much time does it take, etc. My suggestion is just have a cold beer afterwards and call it even.

If you have no other alternative, log it as walking at an appropriate pace for the amount of time you did it, erring on the slower side. Yes, if your mower is not self-propelled, you'll possibly be under-estimating the calories. I wouldn't worry about it. Yes, if you don't carry your phone, you'll be guessing at the pace based on your past walking experience. But this approach would give you a few extra calories burned without a big risk of over-estimating them.

Another alternative, if this is something you do regularly, is not to log it as exercise at all, and treat it as part of your activity level setting (monitor your weight loss rate, if you lose weight at the expected rate over time, your activity setting is reasonable). That's probably what I'd do, for some routine life activity I did on a semi-predictable schedule that wasn't very intense.

If you're at a large calorie deficit (2+ pounds lost per week), and are usually very, very sedentary, I can see why you might want to log something and not further increase your calorie deficit. But if you're losing more slowly anyway, not eating back lawn-mowing calories probably isn't injurious.