My Woods 59202 Lamp Timer doesn't have a switch labelled "ON/AUTO" as per the instructions, but obviously the only switch on it, labelled "ON/OFF" is the manual override. The problem is that since "OFF" must mean "AUTO" then the timer appears to not be set according to the included schedule as it should have the lamp on now, but it doesn't. Am I missing something?

Open following link to identify timer and download possible manual:
http://waterheatertimer.org/Woods-timers-and-manuals.html#troubleshoot
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http://waterheatertimer.org/Woods-timers-and-manuals.html#50009
http://waterheatertimer.org/Intermatic-timers-and-manuals.html
http://waterheatertimer.org/GE-timers-and-manuals.html
Add a comment and describe timer in detail, including brand, buttons.
Attach image of timer.

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If Woods 59028 timer turns on-off, then it is probably wired correctly, assuming white wire is connected to neutral or ground as shown on following link.
http://waterheatertimer.org/Woods-timers-and-manuals.html#59028

Lights turn on-off with override button, so timer is probably not beyond rated capability.
That leads back to programming.
Before beginning, always reset timers to clear out possible conflicts.
Next, program carefully so there are no programming conflicts or overlaps,
Make sure that each ON has corresponding OFF, and that selected day(s) match for both ON and OFF.
Woods 59208 timer has 1 year warranty for defects in workmanship.
Buy timers at local box store so returns are easy ... you can buy new timer and return defective timer in new package long after warranty period expires ... giving yourself free extended warranty.
About 4% of new timers arrive defective.

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Woods 59202 is 7-day analog VACATION timer and cannot be set.
Vacation timers are preset to turn on-and-off at random times so it looks like somebody is home when you are away.
http://waterheatertimer.org/Woods-timers-and-manuals.html#59202

You might want the 59201 analog timer which can be set for on-off times over 24-hour period. Minimum on-off is 15 minutes.
http://www.amazon.com/Woods-59201-Appliance-Grounded-3-Prong/dp/B001Q9EFTQ

If you need a 7-day timer, buy a programmable instead of analog. Wooods 59377 or Intermatic DT630 heavy duty
http://www.amazon.com/Woods-59377-Digital-Appliance-Settings/dp/B000IKQRTU

If you have analog plug-in timer, open following link to see how to set timers:
http://waterheatertimer.org/How-to-set-timers.html

Open following link to identify timer
http://waterheatertimer.org/Woods-timers-and-manuals.html#59377

59377 manual
http://waterheatertimer.org/pdf/Woods-59377.pdf

Auto: timer must be set in AUTO for programming to work
The word AUTO is printed on timer face.
Small black triangle will display above the word AUTO when timer is in AUTO

Countdown is feature that lets you set timer ON for set amount of time, and when time expires, timer turns OFF.

Manual button.
I have not tested this specific timer.
Simply plug lamp into timer, and then push manual button to explore feature.
Watch what symbols display, and watch what lamp does.

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You posted under Woods vacation timer:
See large image

This is a 7-day timer.
Days are printed on the dial.
Each day has night time with dark background and daytime in light background

If you want to set time for Monday at 12 noon, rotate dial clockwise until Monday 12 (in light background) line up with arrowhead located on inner dial.
After that, move switch to TIMER

Timer turns lights ON-and-OFF at random times throughout week

Woods TN300 and Intermatic TN300 and Woods 59200 are same timer:
To confirm this is your timer, open link below
http://waterheatertimer.org/Woods-timers-and-manuals.html#59200
If this is not your timer, then add a comment for additional free service.

Instructions:
Rotate dial to set current time, when plugged in, dial rotates
48 segments located on dial; each segment is 1/2 hour. Push segment down like piano key.
Segment pulled up = lights OFF
Segment down = lights ON
Override switch on top of timer.
For programs to operate, move switch to Timer ON

Plug lamp into timer.
Turn lamp ON
Plug timer into wall

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Intermatic TN300 timer is same as Woods 59200 timer
Timers are sold under different brands and numbers depending on who is packaging and marketing the product.
59300 manual:
http://waterheatertimer.org/pdf/Woods-59200.pdf

Rotate dial to set current time, when plugged in, dial rotates
48 segments located on dial; each segment is 1/2 hour. Push segment down like piano key.
Segment pulled up = lights OFF
Segment down = lights ON
Override switch on top of timer.
For programs to operate, move switch to Timer ON

Plug lamp into timer.
Turn lamp ON
Plug timer into wall

If instructions above don't match your timer, see following page to identify timer:
http://waterheatertimer.org/Woods-timers-and-manuals.html

59419 is the Woods 2 pack plug in 24-hour dial timer with 2-prong outlet.
If this is not your timer, add a comment.
http://www.amazon.com/59419-Appliance-2-Prong-Outlet-24-Hour/dp/B002Y1KZHQ

See all Woods timers and manuals:
http://waterheatertimer.org/Woods-timers-and-manuals.html

Manual for Woods 59419:
http://waterheatertimer.org/pdf/Woods-59419.pdf

Instructions:
Rotate dial until current time lines up with arrowhead located on inner dial
When plugged in, dial rotates
24 segments located around outer edge of dial; each segment is 1 hour
You have to pull these segments up using fingernail or small screwdriver
Segment pulled up = lights OFF
Segment down = lights ON
Override switch located top of timer, move switch to TIMER
Plug lamp into timer
Make sure lamp is turned ON
Plug timer into wall

Try the links below to set the timer.

http://waterheatertimer.org/pdf/Woods-59201.pdf

http://waterheatertimer.org/pdf/Woods-59202.pdf

http://waterheatertimer.org/Woods-timers-and-manuals.html

See this website for links to instruction sheets in PDF format for Woods US timers:

http://www.woods.com/us-timer-instructions.asp