What is if someone took a neder on himself, and said eg. "i will not eat potatos anymore," but his intention was only for a certain Kind of potatos. Are now all potatos forbidden to him (as the person said) or only that certain kind of potatos (as he intended while saying "potatos")?
The Shulchan Arukh in the laws of Nedarim (YD 210:1) rules that if one intended to prohibit wheat bread to himself as a Neder but only said "bread" unmodified, only wheat bread is prohibited to him. (The Shakh there rules this is only so if the omission of the type of bread was an accident, and one cannot rely on this rule Lechatchila.)
Your case ("i will not eat potatos anymore") is a Shevua (an obligation on the speaker, not a prohibition on the item), and the Shulchan Arukh in the laws of Shevuot (YD 239:1) rules that Shevuot have the same rules as Nedarim in this regard.
So in your case (assuming it was an accidental omission) only the specific kind of potato would be prohibited.
The answer to your question is that if you are an am haaretz (which would include anyone who doesn't know the halachos of nedarim), then we don't care that you meant a specific type of potato in your mind and you are assur to every potato since there are so many different types (at least 7: russet, red, white, yellow, blue/purple, fingerling and petite), we say that you are assur to **all potatoes**.
Shulchan oruch Yore dea 208,2:קונם אשתי נהנית לי ואמר לא נדרתי אלא מאשתי הראשונה שגרשתי אם הוא תלמיד חכם נאמן ואינו צריך התרה ואם הוא עם הארץ צריך שאלה שלא ינהגו קלות ראש בנדרים - if someone says "A vow that my wife can't have benefit from me," and then he explains that he meant his first wife that had been previously married to me, If he is a Talmud chochom he is believed and doesn't need a hatoro (nullification from 3 people), but if he is a worldly man he needs nullification so that he doen't act irresponsibly with nedarim.
The exception is when he says "bread" where everyone knows that he meant only bread made of wheat flour since this is what type of bread that the vast majority of people refer to as bread stam without explanation. see S.A Y.D 210,1.היה בלבו לידור מפת חטים והוציא בשפתיו פת שעורים מותר בשניהם אבל אם היה בלבו פת חטים הוציא בשפתיו פת סתם אסור בשל חטים לבד
And the beis Yosef explains in the name of the Rashba The reason is wheat flour bread is the type of bread that the vast majority of people refer to as bread stam and not barley bread - ונראה דדוקא בגמר בלבו להוציא פת חטין והוציא פת סתם הוא דאמרינן שלא מיתסר אלא בפת חטין משום דסתם פת דחטין הוה אבל גמר בלבו להוציא פת שעורין והוציא פת סתם לא אמרינן דלא מיתסר אלא בפת שעורין דהא סתם פת לאו דשעורין הוא
Even the Rema who says if you think of barley bread and say bread stam, the neder/ shavua works, he comments in the Darkei Moshe 210,1 מבואר דפת שעורין נמי פת סתם מקרי ודינו כשל חטים - this is Because barley bread is also commonly referred to as "bread" stam so that's why it has the same halacha as wheat bread.
A random breed of potato according to Shulchan aruch and rema is not the standard potato as There are more than 200 varieties of potatoes sold throughout the United States. Each of these varieties fit into one of seven potato type categories: russet, red, white, yellow, blue/purple, fingerling and petite. There are so many different types of potatoes that people use, one cannot divine the species intended by the Noder, unlike bread, where in general one would either be referring to only wheat or barley. Therefore you would be prohibited on all potatoes!
at least you'll loose a few pounds..