..the form asks for all visited countries
This is as clear as can be - You Should write all the countries you have visited ... Schengen or not . Don't over-think this. Countries are Countries. Treaties and Conventions are treaties and Conventions.
It is true that inside the Schengen area almost no one can know - but if you have nothing to hide just list them all.. It is actually usually an advantage when they see a seasoned traveler .
In the last application I filled ( just 3 days ago ) I had to list all the countries in the last 5 years ! ( Total number : 60 )
Does it matter much if I forgot to enter a country because I don't
remember going there?
The important thing is not to lie . If you do not remember is one thing. But if you omit and they find a stamp on your passport is another .
One of my Friends once made a mistake and listed
DPRK ( Democratic People's Republic of Korea, or better known as North Korea) instead of
ROK ( Republic of Korea - or Korea ) on his US visa application ( not less... ).
He got confused by the " Democratic " part - but When he got into his interview he just said he was mistaken, showed the SOUTH stamp in this passport - and all was OK ...
Does the applicant's nationality make a difference?
Of courses it makes a difference . In some cases - even the nationality of the FATHER or MOTHER makes a difference . ( India Visa application form has a specific field to certify that parents are not Pakistan nationals, nor were born there . Pakistan has the same for India ...)
..Or does it entirely depend on the country issuing the visa?
Yes, different countries looking for different things.
Some look for Security reasons, Some for health Issues , Most for both . Don't forget political issues . ( Pakistan-India , Israel- Arab countries , China - Taiwan / Tibet ).
Use your common sense . If you Apply for a Chinese visa - don't list Tibet. If you apply for a US visa , try not to have Iran or Syria on your list .. If you apply for Iran visa, Don't list Israel.
Many such political-issues-affected Countries are sensitive and understanding of this issue, and for example, Israel boarder control allow(ed) you to ask them to NOT STAMP your passport on entry. ( today they do not stamp anyone anymore . You get an e-entry ticket)
Until not so long ago, When a Chinese citizen was entering Taiwan ( As Single traveler - not a group ) - they were issued a special "passport" or Document that was validated at the border and on the exit they should have returned it . That way they would not have any stamp on their re-entry to China and no one could have known they have visited there . This practice to my knowledge is deprecated now that thee relations between China and Taiwan has warmed a bit and some provinces residents can travel freely to Taiwan.
I apply for almost 20 different visas a year , and after so many applications I can tell you that this field ( visited countries) is the least problematic one ( except for obvious cases of Ebola stricken countries and political enemies )