The explaination given by you is not completely correct. I am reproducing the Rules for Waitlisted e-ticket passengers from an e-ticket:
Rules for Waitlisted E tickets passengers
a) Status of E-tickets after chart preparation
1. Confirmed E ticket –Wait-listed E- ticket where all passengers are confirmed at the time of charting.
2....
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Partially waitlisted/Confirmed/RAC E ticket - Wait-listed E- ticket where some passengers are confirmed/RAC and others are wait-listed at the time of charting.
3. Fully waitlisted E tickets - Wait-listed E- ticket where all passengers continue to be waitlisted at the time of charting.

b) Authorization to board the train
1. Passengers with Confirmed E ticket are permitted to board the train. Their names will appear on the chart.
2. Names of passengers with Partially Waitlisted /Confirmed/RAC will appear on the chart (including the Waitlisted passengers).
3. PNRs having Fully waitlisted status will be dropped and names of the passengers will not appear on the chart. They are not allowed to board the train. If full waitlisted passengers are found travelling, they will be treated as without ticket and charged as per extant Railway rules.
c) Cancellation rules
1. Confirmed E ticket can be cancelled by passengers through internet before chart preparation only.
2. Partially waitlisted/Confirmed/RAC E ticket can be cancelled by passengers through internet before chart preparation only.
3. Fully waitlisted E tickets can be cancelled by passenger using internet before chart preparation. After chart preparation it will be cancelled by Railways and money refunded automatically.

d) Refund rules
1. After chart preparation, Refunds for Confirmed/RAC/Partially Waitlisted E ticket will be processed through TDR process. Customer can file TDR online and refund sanctioned by Railways would be credited back to user/agents account.
2. After chart preparation, Fully waitlisted E ticket will be cancelled by Railways and refund shall be credited to user/agents account automatically.
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So in case of partially wailtist ticket, passengers name will appear on chart and passenger is allowed to board train.. It is not cancelled automatically in case of e-ticket. One has to file TDR for getting refund.