The LM notation refers to Aurifeuillian factors, which are basically non-obvious algebraic factors. In the case of 2LM, if n can be written as 4k-2, then 2^n+1 factors as L = 2^(2k-1) - 2^k + 1 times M = 2^(2k-1) + 2^k + 1. So, for our case, 2,2214L, 2214 = 4 * 554 - 2. Therefore, it can be written as the product of L = 2^1107 - 2^554 + 1 and M = 2^1107 + 2^554 + 1. The number we factored is a cofactor of the former, hence 2,2214L. The complete list of Aurifeuillian factors for the Cunningham tables can be found here.

Actually, in the case of 2,2214L, there was a better way to write it. By writing 2^2214+1 as 2^18 x^36 + 1 where x = 2^61, it can be factored algebraically, producing among others the smaller, easier to factor number, 64 x^12 - 32 x^9 + 8 x^6 - 4 x^3 + 1. This is what we actually used.