The tester you link to is a continuity tester, not a voltage tester. It states that "NB Not suitable for direct connection with live circuitry". I do not think any results you obtain with this are valid - you need better tools. If it behaves as the voltage test screwdrivers, it doesn't enable you to measure the potential between two chosen points, only between yourself and a point.
I understand that the motor doesn't run, which indicates an issue, but it may be hard to work out what that is.
The most common fault for seeing a voltage on neutral is a broken neutral. Capacitive coupling, combined with the high input impedance (MOhms) for most meters results in voltage showing. This will generally not be constant and will visibly change once the meter is connected. These are often termed "stray" or "ghost" voltages. Fluke has some information on this. The solution is to present a low impedance load in the range of kOhms. Some multimeters have this built in as a Lo-Z (Z for impedance) function - the Fluke 289 and 117 are ones that do.
Russell's lightbulb tester is essentially performing the same task - it draws current to light the bulb so won't respond to stray voltages.
A broken neutral would also stop the motor running - this is the first thing I would check.
Have you checked if any other appliances work in these sockets?
UPDATE - based on the lightbulb tester not lighting across any pair, but lighting between phase and an arbitrary earthed point, it sounds like both earth and neutral may be disconnected. This is an uncommon fault across an entire building in Europe as nearly all buildings use a TN-S earthing system where neutral and earth are delivered as discrete conductors from the substation. It may be that TN-C or TN-C-S is used in your area, where a fault in the cable supplying the building can easily break both neutral and earth. Refer to the wikipedia page on earthing systems to see diagrams.
Of course, neutral may be floating and there might not be any earth connected at all - stranger things have happened.
For diagnosing mains issues around the home, I always recommend two pieces of test equipment. Your profile says India, and I am not familiar with availability of test equipment there, but in the UK:
- A simple voltage only tester such as a Fluke T50, which costs approximately £25. These only measure voltage and are very safe as a result. I have seen several incidents now where over-enthusiastic DIYers have either tried to "see if 13A was coming from the socket" or left the probes in current mistakenly. You can diagnose most common issues without needing to measure current.
- A socket tester - these detect a number of conditions such as phase/neutral swapped, earth missing etc. Very quick and easy to use.