The other piece of advice is trickier, and I personally do not follow it too closely (everything in life is circumstantial:). Two aspects to mention:
1. Often, a lie is more harmful than a slightly harmful truth. And opportunities to get caught in a lie start with the application process - some applications have multiple seemingly unrelated or different questions that aim to reinforce the validity of your claims; this will also sometimes be reinforced with interviews, reference checks, etc. And then if you get the job or grant or whatever, there's still the risk of getting caught at virtually any point in the future.
My wife and a few of her friends have been HR managers at quite varied corporations, and universally they lament that people get fired over an insignificant lie. What they lied about might've been a verbal "Hey don't do that" or a formal reprimand, but lie got them immediately fired.
2. Variation of that but, perfection and/or fakeness can stand out. Not to say there aren't people who can lie/fake perfectly, and sometimes many of us think that we can pull it off better than we can. But while I cannot claim that I have never fibbed or concealed in my life, last few decades I've been lucky enough that I didn't need to.
And luck is an important word; I've been lucky professionally since I came to Canada, which enabled me to have good success going counter to that advice: e.g. to every new manager, I proactively indicate explicitly that I "Attended university but have not graduated", I came extremely upfront when I started photography business even though it had nothing to do with my IT dayjob, etc. I find one's experience at large companies is partially shaped by formal policies written by people far away from you, but also hugely by the actual people surrounding you, and I've been lucky / chosen well over the years. Milleage most definitely WILL vary, and I've been in sufficiently different / more precarious or dangerous situations to be fairly aware of my current privilege.