If we take the same example and refer 3"rd person. Than you might be happy.

Bilgisayarımı kaybetmesini istemiyorum.

I don´t want him to lose my computer.

[literally, I don´t want his losing my computer]

I know I am very stubborn (what can I do, I am researcher..), but when you use one or another..; I saw written both of them; when I ttranslate in my language I cant see the difference but i am sure there is one..or not...

to be or not to be...

kusura bakma...

Being stubborn in learning is a good quality. Bravo sana ! 

One of my old post due to this issue ;

The verb istemek can take either  "infinitive verb" or " verb in accusative form" before it.

Oraya gelmek istiyorum ---> I´d like to come over there. --> Infinitive form

Oraya gelmeyi istiyorum ---> I´d like coming over there. ---> Accusative form

If you like to know the differences between two sentences then look at the "red highlighted" letters. In practice we Turks would use both for the same situation. 

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Türkçe öğrenmek istiyorum ---> I´d like to learn Turkish

Türkçe öğrenmeyi istiyorum ---> I´d like learning Turkish.

In both forms the first verb[al] [öğrenmek and öğrenmeyi ]  can be classified as an object of the main verb [istiyorum]

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However, using the accusative case for the verb before´istemek´ is NOT always sounding good in Turkish , such as ;

Çay içmek istiyorum ---> I´d like to drink tea. ---> This is what we would normally say.

Çay içmeyi istiyorum ---> I´d like drinking tea.--> This form is sounding bit weird.

May be it is like in English, you wouldnt say "I´d like [to] drinking tea" instead you normally would say " I´d like to drink tea"

so.it is about the action in course...

if this action means a movement (somehow, like drink, eat) you use -mek/mak

and

if is about non movement acion (like learning, teaching) you  use accusative form...

what do you mean by non-movement action ?

how can I say....

when you are drinking you take the cup (of tea) from table and bring it to mouth -

but  when you are learning something you dont need to do any movement (the action is only of your brain)

something like this...