This depends on the type of laptop and power supply, some have additional logic signals, etc.
In all likelihood, your lab power supply is not able to provide enough current to power your laptop. Laptops use a good amount of power, if you have the original power supply it came with, you would see a current rating in mA or A. This is the total amperage the power supply can provide, and my guess is that it is at the least, 2 Amps or more. You may be able to charge the battery with your lab supply though.
There are other possibilities too, but without knowing the laptop model and the specs on your lab power supply, I couldn't say. Your lab supply may very well be able to power the laptop, however when the laptop is first turned on, your power supply may not be able to quickly adapt to the load.
the power supply is able to provide 3 Amps and the Current measurement
shows that consume doesn't exceed 1.5 Amps
There are a lot of possibilities here, perhaps your lab supply isn't accurate or there were power fluctuations. Maybe your adapter plug was not the correct size, so you had a poor or intermittent connection. The laptop may have had an unrelated issue, that caused this problem, I've seen some that turn off as soon as you plug in external power. Perhaps the battery was bad, problem with the motherboard, or maybe a fan wasn't working. There are just a ton of possibilities.
the laptop is an old lenovo...
"the beep and dead" seemed to be a power security act...
HOW can the laptop distinguish between the its adapter
Usually a beep like that is a bios problem or warning. Also, according to this, it seems that Lenovo has different power supplies and depending on what one is used, it will throttle it's performance. A power supply for a laptop doesn't just have to provide power, sometimes there is some data communication/feed back, perhaps even just a way to verify that the power supply is a real brand name one.