The short version of my answer is yes, I believe it's safe to say that temptatio and temptation with respect to its modern secular usage can be, for the most part, considered false friends. However, before I get to that, consider the meaning of the word which was translated as temptation in the passage in question, i.e., Matthew 6:13.
The word tentationem which appears in the Vulgata for this verse is the Latin translation of the Greek word πειρασμόν. As @brianpck said in his answer, This comes from the verb πειράζω: to "try, tempt, test." According to Vincent's Word Studies, the meaning of πειρασμόν in this particular passage is:
Temptation [πειρασμον]. It is a mistake to define this word as only
solicitation to evil. It means trial of any kind, without
reference to its moral quality. (emphasis added)
When this meaning is compared with the definition of temptation in the modern usage, the idea of a test or trial is usually absent. In the Cambridge Dictionary, for example, there is no mention of either a test or a trial. Among its various definitions, the following is a good example of how it's often used in the modern sense:
the desire to have or do something, esp. something wrong, or something
that causes this desire
However, this was not always the case. In the 1828 version of Webster's dictionary, we find the following:
The act of tempting; enticement to evil by arguments, by flattery, or
by the offer of some real or apparent good.
Solicitation of the passions; enticements to evil proceeding from the prospect of pleasure or advantage.
The state of being tempted or enticed to evil. When by human weakness you are led into temptation resort to prayer for relief.
Trial.
Lead us not into temptation
Of particular interest is the fact that Matthew 6:13 is quoted in the dictionary as an example of the usage of temptation in the sense of a trial.
What I would conclude from this is that temptation and tentatio weren't originally false friends, but rather a semantic shift occurred over time such that the modern usage of the word has, for the most part, lost the meaning which was inherited from the Latin tentatio (i.e. an attack; an attempt, trial, proof).
Although I didn't hear what the Pope had to say, my guess is that he was defending the original meaning against possible misunderstandings caused by the modern usage.
This leads to the consideration that, besides the etymological basis of πειρασμός and the particular context, the meaning must be understood according to it consistency within the logical structure of theology as a whole. In other words, an improper understanding of its meaning would lead to contradictory interpretations.
It should also be observed that πειρασμός is used elsewhere in the Bible in different senses, including the negative sense of enticement to sin, which would probably explain why the word temptation is used as it is in modern times.
A good example of these two points (i.e. logical consistency and diverse senses) is James 1:13:
No one is to say when he is tempted, “I am being tempted by God”; for
God cannot be tempted by evil, and He Himself does not tempt anyone.
This verse presents an apparent contradiction with Genesis 22:1 where we find, "Now it came about after these things, that God tested Abraham…" (In the LXX: ἐπείρασε, aorist active indicative of πειράζω; translated as "tempted" in the KJV). Of course, this difficulty is easily resolved when it's understood that James is using the negative sense of the word. That is, God puts man to the test, but He is never to be accused of actively enticing him to evil. Dan G. McCartney explains:
The abilities to tempt and to be tempted are rooted in an evil
capacity within the person. It is this capacity within that is of
concern to James in 1:14. […] Things start with 'one's own desire (τῆς
ἰδίας ἐπιθυμίας, tēs idias epithymias), which places responsibility
firmly on the shoulders of the individual.
This is consistent with a comment made by Augustine:
Hoc enim peccati nomine appellat [Paulus], unde oriuntur cuncta
peccata, id est, ex carnali concupiscentia.
For Paul gives the name of sin to that from which all sins arise,
namely, the lust of the flesh. (Sermo CLV, in reference to Romans 7)
The proper sense of πειρασμός is also theologically important in respect to understanding in what sense Jesus was said to be tempted. This is explained very well by Daniel Schrock:
We must conclude instead that [Jesus'] experience of πειρασμός or
"temptation" is more akin to what we refer to when we speak of
"testing," which is another common, yet overlapping meaning of the
"temptation" word group in Greek. That is to say, Jesus' experience of
"temptation" is a "testing" of what he will do. It is a trial into
which he is placed in order for him to demonstrate his obedience. It
is not by any means an experience in which he is inclined and
internally lured by illicit desire. Rather, it is a moment in which
his impeccable purity of heart is manifest and proven by means of
external trial.
Answers to your specific questions:
Question 1:
Shouldn't the language in the title be translated as something like
"Do not put us to the test"?
That would be an acceptable translation. However, as noted above, the word temptation was originally understood in a sense that was suited very well to the meaning of the text.
Question 2:
Are there any citations where temptatio, πειρᾰσμός, or the related
verbs are used to convey the nuance of inducement, seduction, or
entrapment that comes with the primary meaning of the word
"temptation" in English?
Yes. As noted above, James doesn't deny that God puts man to the test, but he does deny (using πειράζει negatively) that any inducement, seduction or entrapment be attributed to God.
In addition to that, Thayer's Greek Lexicon gives other instances of such usage:
c. to try or test one's faith, virtue, character, by enticement to sin; hence, according to the context equivalent to to solicit to
sin, to tempt: James 1:13; Galatians 6:1; Revelation 2:10; of the
temptations of the devil, Matthew 4:1, 3; Mark 1:13; Luke 4:2; 1
Corinthians 7:5; 1 Thessalonians 3:5; hence, ὁ πειράζων, a
substantive, Vulg. tentator, etc., the tempter: Matthew 4:3; 1
Thessalonians 3:5.
Question 3:
Similarly, are they any cases of the word "temptation" in Modern
English being used unequivocally to mean "putting something to the
test"?
According to the Cambridge Dictionary, such usage isn't common. However, there are such cases among those who understand the Bible. As quoted above, Daniel Schrock unequivocally uses it in that sense:
That is to say, Jesus' experience of "temptation" is a "testing" of
what he will do. It is a trial into which he is placed in order for
him to demonstrate his obedience.
Question 4:
In short, aren't "temptation" and temptatio really false friends?
Yes and no: yes, I believe that that would be a fair assessment with respect to modern secular usage (as noted above). But, no (at least as it is used in the Bible) with respect to the more archaic usage of temptation, as well as in certain theological contexts.
