What is the name of a chord that contains the following intervals: 1, 2,3,5?

TU to David.

A little more detail: While in common practice sus2 and add2 are often pretty much the same thing, sus does mean that there is a suspended note instead of just an added tension tone.

+2 is just bad notation (and there's plenty of it out there.) + means augmented triad, which has nothing to do with the 2nd.

So to answer your question, the notation should reflect the intention of the composer and be interpreted easily by the reader. If the composer intends a preparation,suspension, and resolution then the chord is a sus, but there's a hidden trick in that answer. If you have 1, 2, 3, 5, what is the 2 a suspension of? Garden variety sus2 would have the 2 suspended in place of the root, which is always tricky because its hard to do that without implying the root anyway. (and in this case you state that chord tone 1 is present.) The only legit use of "sus2" notation would likely be a highly chromatic harmony in which there's no root OR sus2 resolves to b2/b9 or gets regressed into a non-root in a completely different chord.

I would say that in 99 percent of all cases where you see 1, 2, 3, 5 the 2 is an add 2. It's a color tone or tension tone that is non-functional (does not have to resolve) or resolves chromatically without actually being a suspension. (because we already have 1 present)

And I'll also say that in the vast majority of all cases where you see sus2 its incorrect notation.

It's a major chord with add2. Could be called a major chord with add9 if the 2nd grade- note is an octave higher.

In both cases you could (or not) add the 7th or minor 7th, but that would change the chord's name to maj7add2 - m7add2. If adding a 9th, then maj9 or m9.

You are correct that it needs the 7th to be called a 9th! You usually don't call it a major triad, though it is, because that is assumed. If you see the chord symbol "C", you assume that is 1,3,5 or C,E,G, If the 2 is also there, it is called a "sus2" or "add 2". Some copyists write it as "+2" but that is too confusing.