I am very new to music theory and I just want justification on what “play 2nd time only” means. Does it mean that the instrument does not play it the 1st time, and then does play it the 2nd time, right?
In short, it means exactly what it says: only play the second time around
Often this is used when a different texture is wanted the second time round, sometimes for the second repeat of a few bars (which will have a repeat sign) or sometimes for a whole verse or the whole piece even (which might not bother to include the repeat signs if it's obvious from context).
3 reasons you might have this:
1) for a verse that has a sparse accompaniment the first time round, and on the second time you want a fuller accompaniment (e.g. starting with just a piano and voice, and on the second time the whole band/the string section comes in too, very common)
2) In jazz arrangements, you often get horn stabs that only come the second time round on a solo section, or that have more instruments the second time round in order to produce a buildup effect
3) In all types of music, you might have a section that repeats, but the second time round there's a lead up into the next section (colloquially often called an "upbeat" although technically this has a slightly more specific meaning). You might find this in Brass band music, big band music, jazz arrangements, theatre music etc. You could write this as a first and second time bar in a repeat, but if the first time bar would be empty and the second one not, often it's easier (and saves space on the page) to just write "second time only" above the notes rather than bothering to have an empty first time bar.
Yes, you're right. This is probably somewhere in the vicinity of a repeat sign:
Terms in music notation are often in Italian, but sometimes plain English is used as well. It needs to be short, because one needs to read it fast while playing the music, so don't look for subtleties; there are none.
This is a common way to avoid writing an entire section of music twice due to one small difference. Say for example you have two verses that are nearly identical but because the melody is slightly different the second time due to a slight lyric difference or there is a small fill the second time your options are to write the entire section twice or write it once, use a repeat, and mark the thing that only happens the second time with a little note to the player.
It is very common that the lead into the next section is different when you repeated so there is a special "second ending" type of repeat that you may encounter as well.