Hard to know whether you are using them all correctly, because they all mean different things. You seem to be using them all in the context of checking that something is correct, and they don't all necessarily mean that.
Testify means to give your personal testimony. A testimony should be truthful and correct, but it is often just a personal perspective. For example, a witness in court may give their testimony, but that alone may not prove or disprove a case - it just adds to a collective body of evidence. A personal perspective can also be flawed.
Justify means to give a reason for. Again this doesn't necessarily mean checking something is correct - you could justify a wrong choice by stating extenuating or mitigating circumstances.
Verify means to make sure or demonstrate that something is true, accurate, or justified. This normally means checking something against another source - for example, if I were to verify the spelling of a word I would do so by checking a dictionary. Your own knowledge can be used for verification.
Certify means to formally confirm that something is correct. It may not necessarily include a paper 'certificate', but that is the level of formality this is normally used for. You are not just saying you think something is correct - you are putting your name to that claim formally.
Looking at your examples then:
- Christopher testifies that "leave home" in the context refers to "set out for the day".
This doesn't seem correct. That isn't his personal testimony. I think it should be "verify"
- Thomas gives a link to justify Christopher's understanding is correct.
Depends what is in the link - if it is a second source that repeats the understanding, then this is verification. If it is some supporting evidence then perhaps you could use 'justify'.
- Anthony verifies Christopher's post and Thomas' link.
Verifying the post seems reasonable, but verifying a link sounds like he has checked the link works, not the content. Perhaps it should say 'citation', not 'link'?
- Cambridge Dictionary certifies that meaning by issuing it in its online dictionary.
I suppose the Dictionary is an authority, but even so, 'verify' would be adequate here. There wasn't really a formal process of Cambridge checking the meaning - you mean that you looked in their dictionary for verification.