There is no one answer. Different couples have different ideas on commitment. Some will consider themselves committed to a relationship very early, others will take a longer time, and some may never commit.

Being single is the opposite of being in a relationship, so definitely not "single". "Committed" is when the partners are committed to each other in a serious relationship with love, honesty, trust, etc. The definition really depends on what the partners agree on, but generally it excludes seeing other people ( = an exclusive relationship). A 1-2 months old relationship, even if it's exclusive, might not be defined as "committed" yet because the partners are still getting to know each other. Once they know each other's needs, plans, wishes and preferences and they are compatible, then they may decide to be "committed", which can then continue as a relationship or even lead to marriage, which is another - official - type of commitment.

I'm not married, but if I told my friends that I was single my girlfriend might be a little bit upset. Particularly as we own a house together. However, on official forms if the only options are "married" and "single" I would put "single". Usually these days they add an option for "partner", "long-term relationship", "long-term partner", or suchlike to avoid this issue.