There probably is confusion between rank and appointment and usage within the army at large during this time.
I can see what you are describing as usage within the regiment, i. e. within the company, exclusive of the rifles and fusiliers, two Lieutenants would be described as the 1st Lt. and the 2nd Lt. of the company. They would all be shown as Lt. in the army lists as the regiment also had Ensigns.
In the Rifles and Fusiliers, the army lists refer to First Lieutenant and Second Lieutenant with different seniority dates for each. For example, in the 23rd Foot, Edward Methold is a First Lieutenant of 23 March 1815 listed above Thomas Lilly as a Second Lieutenant of 1 October 1812. They were probably addressed as Lieutenant without distinction, except perhaps for official correspondence.
The Royal Artillery had 2nd Lieutenants [who received 4s. a day in 1793.]
Then of course we have the Lieutenant Colonels in a regiment where they are variously referred to as senior, junior, 2nd, etc. There is no such distinction in the army lists. By common usage, they are addressed as Colonel.
Next is Brigadier or Brigadier General [both are used interchangeably] which is a rank associated with an appointment. It is not a substantive rank at the time. The last appointments to substantive rank were in 1746 as far as I can tell. Confusingly, any officer who commanded a brigade is also sometimes referred to as its Brigadier, regardless of his actual rank, be it Lieutenant Colonel, Colonel or Major General.
Then of course, the Major Generals and Lieutenant Generals are all addressed as General, even though there is a rank of General in the army.
Let's not even touch the Guards....