Looking for a word to describe telling somebody only part of something because of insecurities

Someone who avoids telling a complete information about themselves out of insecurity sounds cagey. Oxford English Dictionary gives one definition, noting that it is colloquial:

colloquial (originally U.S.). Not forthcoming, reticent, wary, non-committal.

One of the examples shows the word used to apply to being non-committal about information related to an event. Someone giving selective information for whatever reason could be called cagey:

1953 G. Heyer Detection Unlimited iii. 31 Aunt Miriam's always a bit cagy about it. What happened?

Here's a list item from an Elite Daily article (Sarah Ellis, "4 Signs Of Avoidant Attachment You Can Spot On A First Date") about dating that captures how the word can be used to describe dating partners who hold back, either due to insecurity or some other emotion:

3. They’re cagey about sharing their feelings.

On a great first date, the conversation should flow easily and naturally. If you feel like they’re hiding things from you, that doesn’t lay the groundwork for a successful relationship down the road.

Perhaps cautious or wary?

Cautious (adj) very careful about what you do or say and unwilling to take risks because you want to avoid possible problems.
[Cambridge English Dictionary]

Example: For once I’m trying to tell you just what it is I truly think and feel instead of the cautious things I told you before.

Or try saying Instead of saying exactly what I think, I’ve been curating my words to protect myself from saying the wrong thing

How about some form of filter? From M-W:

filter: something that has the effect of a filter (as by holding back elements or modifying the appearance of something)

filter: to subject to the action of a filter

The first person was filtering their communications with the second person to avoid potential upset. The first person is now going to communicate unfiltered, i.e., the first person is now going to communicate with the second person without the "potential upset" filter.

Also, if you are interested in idioms, the first person is no longer going to walk on eggshells. From The Free Dictionary:

walk on eggshells: To act with great care and consideration so as not to upset someone.

Cagey has already been mentioned.

In addition to that, there are:

discreet - Having or showing discernment or good judgment in conduct and especially in speech, esp. capable of preserving prudent silence.

guarded - On one's guard; esp. (of speech, behaviour, etc.) careful, cautious, unrevealing, non-committal. [source - SOED]

Try pussyfoot; it means “to avoid making a decision or expressing an opinion because you are uncertain or frightened about doing so”. I think this fits what you’re trying to say but I’m not sure how common this word is nor have I seen it used in a formal context. Also, the definition doesn’t imply reluctance to speak the truth due to someone else’s insecurities but that’s how I usually see it used. Another term is sugarcoat and I think this one is better and more known. The phrase “sugarcoat the truth” is used a lot. Anyway to sugarcoat means to use soft language to make something sound less harsh or more pleasant than it is. https://dictionary.cambridge.org/us/dictionary/english/pussyfooting