Your mother's first cousin is always your first cousin, once removed. That is at the core of what first cousins and removed cousins are. That relationship will never change; your mother's first cousin will always be your first cousin, once removed.
The term "cousin by marriage" is relatively new and undefined, and for many it really does not mean anything. Some people use the term to refer to someone who has married a cousin. In that sense, if someone marries your "first cousin once removed," that person would be your "first cousin once removed by marriage."
It is completely incorrect to claim that someone who marries your first cousin is your "first cousin once removed" simply because of that marriage.
Your mother's first cousin is YOUR first cousin, once removed. The child of your mother's first cousin is YOUR second cousin.
Your mother's aunt's son is her first cousin and your first cousin, once removed.
First cousin, once removed.
If your mother's cousin is her first cousin (if one of each of their parents were siblings) then that cousin is your 'first cousin once removed'. If that cousin has any children, they are your second cousins.
Your mother's first cousin's child is your second cousin. To your children this person is a second cousin once removed.
Your first cousin's son is your first cousin once removed. In many places a marriage between such people is accepted, in others it is not.
Whether you can marry your first cousin twice removed depends in part on the laws of the place where you propose to have the marriage performed. In most of the US, you can marry your first cousin, so marrying your first cousin twice removed would also be permitted.
Your mothers niece is your first cousin. Your first cousin's child is your first cousin, once removed.
The child of your first cousin, once removed, is your first cousin, twice removed, if your first cousin once removed is the child of your first cousin. If your first cousin once removed in the first cousin of one of your parents, the child is your second cousin.
Your mother's first cousin in your first cousin, once removed. Her second cousin is your second cousin, once removed.Your mother's first cousin, once removed could be your first cousin twice removed or your second cousin, depending on the relative positions of your mother and the cousin in their family tree,Your first cousin once removed.
If your first cousin once removed is your parent's first cousin, then your first cousin once removed is your child's first cousin twice removed.If your first cousin once removed is the child of your first cousin, then your first cousin once removed is your child's second cousin.
Your parent's first cousin is your first cousin, once removed. Your parent's first cousin, once removed is your first cousin, twice removed. Your parent's second cousin is your second cousin, once removed. Your parent's second cousin, once removed is your second cousin, twice removed.