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In contemporary English the word "wife" by itself carries the meaning of an official legal wife, and no other modifier or adjective is needed. If you wanted to be explicit you could say "legal wife", but it's really not necessary. The only time I'd expect to hear it is discussions of polygamous or polyamorous families where only one woman can be considered the legal wife by law.

In Anglo culture there is the idea of a trial period, but it takes place before the wedding. We call this the "engagement" period, and the couple are call each other their fiancé (male)/fiancée (female). There is no concept of a trial period after a legal wedding, so there is no terminology to use for such an arrangement.

Partly jokingly, we could say 'lawful wedded wife', which was a phrase used in the marriage service of the Church of England's Book of Common Prayer and possibly other churches. (Some people very jokingly say 'awful wedded wife'!)

It looks like you are inventing a type of marriage. There isn't - and cannot be - a 'usual term' to describe something that you have invented.

In Australia, de facto is a term used for what seems the opposite of your situation, namely where people are carrying on as if married (including mortgages, kids etc), but haven't been to a registry/ceremony to officially tie the knot. In this case she would be the de facto wife and he, the de facto husband.

You also mention in comments that this is for a story. Maybe the worldbuilders could come up with some suggestions for you. In that vein, I would consider making up a term to go with your invented scenario. Something like 'testwedded' or 'maybetrothed'.

A is either someone's wife or she is not. The term husband or wife, used alone, implies officiality. If a couple are merely living together without having gone through a form of legal marriage, each may be called the "common law" husband or wife of the other. I suppose you could call her his "legal wife", but it would sound odd to Western ears. As Ben Voigt notes: being officially married is a matter of legal fact, and does not depend on a man's opinion (what he "considers").

Not quite English, but in the Philippines that would be “The Legal Wife”. Also the name of a TV series.

Stick with "Husband" and "Wife" for the full, legally married terms.

However, the "pre-marriage contract" you are attempting to (re)invent for your story is a betrothal. The people thus joined are each others' "betrothed".

A betrothal was a semi-binding contract, typically with "exit" clauses for both parties. Sometimes - typically among nobility - a betrothal contract formed part of a joining of families or businesses (e.g. King A turns to King B: "I say, old chap. In a spot of bother here, minor disagreement with King C. Could use some help putting the blighter back in his place. If you were willing to lend an army or two... Well, your son's eye has been roving, and my daughter is of marriageable age now. What do you say, dear chum?")

As for your "trial marriage" aspect?

In some historical cultures (including colonial North America), the betrothal was essentially a trial marriage, with marriage only being required in cases of conception of a child.
_Wikipedia

The two are already married on paper.

Btw. the reverse case happens all the time. People get married without going to the registry office - they have a big party, perform the required rites at their local place of worship and then they are considered husband and wife for all purposes that matter to their community. It is just in places with a sufficiently strong bureaucratic tradition that a piece of paper with an official stamp is more important than anything else.

In legal contexts, one can say: legal wife versus common-law wife.

• But I am his legal wife. [a character might say]

That means I married him under the law.

In a dialogue, here, we'd say legally married or legal wife.

Civil registry in AmE is vital records office. Civil registry is used in English sometimes but it is inevitably a translation from French or Spanish, etc.

A: That's not something to call [...] when I'm his wife, legally. or: when I'm legally his wife.

B: But maybe he doesn't see you as his legal wife yet. or: as legally married yet.

If you are married, but not together anymore (either temporarily or permanently), then you are separated.

See for example Different Types of Separation: Trial, Permanent, and Legal Separation and also legal separation.

Disclaimer

Attention: My answer is not officially recognized in any textbook or by any english language scholars.

I admit that my answer manipulates the rules of english to conform to a specific style of writing used by very small group of authors and scholars.

If you use my answer in any other context besides creative writing or poetry, your usage will be judged incorrect.

You have been warned!

However, despite my warnings, do not be afraid to hack the rules of the english language to enhance communication with your audience.

Answer

For your specific question, I suggest that you capitalize the word wife to indicate the meaning has transitioned from a common noun into a title or archetype, i.e. Wife.

I recommend Wife to subtly emphasize her focus on being addressed in the proper manner because she is not his wife but the ruler of his heart.

She believes she is not a queen but the Queen of his heart.

Therefore, her role is not that of his wife but his Wife.

The other character corrects her misconception that they exchanged something of greater value than what is written in their marriage license.

A: This is not something to call [...] when I'm his Wife.

B: You may be lawfully wed to him, but no law can force his heart to contract that you are his Wife.

You could summarize this subtlety as a footnote for readability to help your readers understand your logic without interrupting the dialogue.

A: This is not something to call [...] when I'm his Wife¹.

B: You may be lawfully wed to him, but no law can force his heart to contract that you are his Wife.

¹ She believes she is not a queen but the Queen of his heart. Therefore, her role is not that of his wife but his Wife.

Thank you for asking such an interesting question!

~ Melioratus