In British parlance the Royal Navy is The Senior Service due to having been created as a permanent establishment in Tudor times, while the Army only became permanent a few centuries later.
The importance of a distinction is the need for military officers to know, at all times, who is the most senior for command purposes. Within each service officers at a specific rank assess their seniority by date of commission, both brevet and substantial. When two services are working together the same applies, except an additional tie breaker is available. - if rank are equivalent and dates of commission are the same, then the Royal Navy officer has seniority over the Army officer.
In 1913 there was as yet no Royal Air Force, and various Corps such as the Royal Artillery and Engineers were simply branches of the Army. So the only available junior service was the Army, unambiguously.
Note that it is possible to look up the effective commission date for all British Officers since 7 November 1665 on the London Gazette Archive. For example, here is the announcement of the promotion to Brevet Field Marshal of Arthur Wellesley in the July 3, 1813 Extraordinary Edition (pp 1270, lower right).
That edition also contains the after action reports filed by Wellington dated June 22 and 24, 1813, following the battle of Battle of Vitoria, including the names of all officers killed or wounded in that engagement.
The Queen's Regulations for the Army 1975 outline in Chapter 2, Parts 3 and 4 and further in Sections 9.163-9.167 the precise workings of how seniority of officers is to be determined:
a. Officers holding substantive rank are to take precedence over all those holding acting or local rank of the same
grade. They are to take precedence among themselves according to their date of promotion to that rank.
J2.041. Under the provisions of the Armed Forces Act 2006, when members of one Service are co-operating with one or both of the other Services they enjoy like power of command over members of another Service as the members of that Service of ‘corresponding rank' insofar as power of command depends upon rank or rate. The expression ‘corresponding rank' is defined in the Act by reference to The Queen's Regulations. Thus when members of different Services are acting together the superior officers of one Service become, in accordance with the scale of corresponding ranks prescribed by these Regulations at the Table
in para J2.042, superior officers of members of the other Services of lower corresponding rank or less seniority and can, consequently, give lawful commands to them.
These principles have remained unchanged over centuries, though details have become more complicated over time.
OED (1928) references:
service, n. 1
b. the service: the Army or Navy (according to the implication in the context) considered as a sphere of duty or occupation, or as a profession. Also, the Air Force and intelligence departments. So the (United) Services, the Army and Navy. For the senior service (the Navy), see senior a. 2.
1706: Farquhar Recruit. Officer iv. i, “Sir, I wou'd qualifie my self for the Service.”
1714: Spect. No. 566 3 “A Man has scarce the Face to make his Court to a Lady, without some Credentials from the Service to recommend him.”
1777: Earl Carlisle in Jesse Selwyn & Contemp. (1844) III. 208 “No domestic reasons can be strong enough to justify a man in quitting the service at the opening of a campaign.”
1833: Marryat P. Simple xxviii, “I wish Mr. Harrison would stay on shore with his wife altogether,—it's really trifling with the service.”
1842: Burn Nav. & Mil. Techn. Fr. Dict. Pref. (1852), “My brother officers, or those of the Sister Service.”
1845: Stocqueler Brit. India (1854) 381 “The merchants and others not connected with `the services' could only be admitted by ballot.”
1862: G. H. Kingsley Sport & Trav. (1900) 362 “Every morning there is a small row of the United Services standing just abaft of the mainmast.”
1872: Routledge's Ev. Boy's Ann. 185/1 “The Service is going to the dogs.”
The Senior Service: the navy as distinguished form the army. [attested from at least 1899, and again in 1911, as below]
senior, a. and n.
- a. That ranks before others in virtue of longer service or tenure of a position; superior to others in standing. the senior service: the navy as distinguished from the army.
1513: Bradshaw St. Werburge i. 2164 “Bycause that Werburge in order was senyoure, Her mother Ermenylde gaue her the sufferaynte.”
1811: Wellington in Gurw. Desp. (1837) VII. 245 “You are aware that he is senior to Marshal Beresford.”
1886: C. E. Pascoe London of To-day xxiv. (ed. 3) 225 “The Inner and Middle Temple..are the two senior Inns.”
1899: Hope Huntly Our Code of Honour xxii, “‘It was my heart's desire in boyhood to enter the senior service’. ‘Then why did you not?’ ’Oh, I yielded to my mother; she was keen on the army.’”
1911: London Mag. Oct. 264 “The Admiral turned round... ‘The Army‘, he said gaily, `comes to the rescue of the senior Service'.”
junior, a. and n.
(b) the lower forms of some fee-paying schools; junior service, the Army; junior stock (see quot. 1914); junior technical school, a school providing a technical and secondary education for boys. […]
1915: E. Wallace Man who bought London viii. 81 “She had a son in the army, and she bore the *junior service a grudge in consequence.”