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.