I have had an interest in the history and etymology of mathematical terms for many years, as witnessed by my MathWords web page. Recently I came across a couple of old journal articles, (1915-1918) related to the history of mathematical induction, and to the term itself. Most of this comes from an article by Cajori, and the very early dates above. Certainly those who know more about the current status of the usage could help by sharing their information.
The logical and scientific process called induction dates back as far as Cicero's translation of Aristotle. Cicero used the latin term "inductio", for the Greek "epagoge", which translates as "leading to."
Induction has always existed in mathematics, but the formal concept of mathematical induction did not appear until it was developed by Maurolycus in 1575 to prove that the sum of the first n odd numbers is n2. While the roots of formal mathematical induction are nested in works from Fermat all the way back, one might say, to Euclid's proof of the infinity of the primes, the work of Maurolycus was unique in the formal use of attaching one term to the next in a general way.
The method of Maurolycus was repeated and extended in the works of Pascal to be a much more clear illustration of the present method but none of them used a particular name for their logical process. Then in his Arithmetica infinitorum in 1656 Wallis decided to name the term. On page 15 he creates the term "per modum inductionis" to prove that the limit of the ratio of the sum of the first n squares to n3 + n2 was 1/3. His inductive method followed very much the unnamed method of Maurolycus.
Later Bernoulli gives an improvement to Wallis' method by showing the argument from n to n+1 as a general proof; this was the real foundation of modern mathematical induction. Bernoulli gives no specific name to his process, but uses his method as an improvement on the "incomplete induction" earlier used.
For the next 150 years, mathematicians used induction in both senses, to refer to the process of observing a relationship from a pattern , and in the method of Bernoulli to prove such an induced relationship by arguing from n to n+1. Then early in the 19th century, George Peacock uses the term "demonstrative induction" in his 1830 Treatise on Alebra. Then several years later, Augustus De Morgan proposes the name "successive induction" but then at the end of the article he talks about the method as "mathematical induction."
Isaac Todhunter used both names in his chapter on the method, but he used only Mathematical Induction in the chapter heading. Several popular textbook authors, Jevons and Ficklin, for example, used both terms. But among several others, Chrystal, Hall and Knight, used only the term mathematical induction. The same name seems to have been common in the early part of the 20th century in America and Europe, with Germany seemingly clinging to a single term for both "complete" and "incomplete" induction. Cajori, in 1918, says the Germans most commonly use the term, "vollstandige Induktion". I do not know if there is currently a more appropriate notation for the true mathematical induction of Bernoulli in Germany. If a reader is familiar with the current situation in German mathematics, please update me.