No. Choosing a special-knowledge or advanced-knowledge basis does not create different family rules. Eligible spouses and unmarried children under 21 may seek L-2 classification through the principal’s qualifying L-1 status.
Separate two evidence questions
The employee’s comparative knowledge belongs in the principal’s case; marriage, birth and identity records belong in the dependent preparation. Eligible L-2 spouses are employment authorised incident to valid status, with suitable documentation, while L-2 children are not authorised to work. The household should plan its move around the actual filing and admission route, regardless of which knowledge argument is used.
The choice of argument does not alter dependent eligibility, which is assessed on the relationship to the principal and the principal's status rather than on the reasoning that supported the petition. What can differ is the practical timeline, since one argument may take longer to evidence than the other, and a household planning around an expected filing date should know which is being pursued. Prepare the family's records in parallel regardless: marriage and birth documentation, certified translations where required, and each person's own travel and identity records.
Note that any work authorization available to a spouse depends on holding and maintaining valid dependent status, and that dependent children cannot work in that status. Keep those records assembled and current rather than waiting for the technical file to be finished first.