Does it matter what type of service of the summons and Dispossessory action is had on the Tenant?

Yes, and no. If the only relief sought by the Landlord is possession of the premises, then personal, sui juris, and tack and mail service is sufficient. If the Landlord is seeking possession of the premises and a money judgment against the Tenant, then you must have personal or sui juris service on the Tenant. If the Landlord seeks to have a money judgment and possession of the premises, the Landlord should note on the return of service "personal or sui juris service," to notify the Sheriff/Constable's department as to the type of service sought.