There are a few separate issues here. First of all, you shouldn't try out various analyses and then choose one to report; if you are fishing around with different analyses that weren't planned ahead of time, you should report everything you did, or at least honestly report that the result you are describing depends on certain analysis decisions (i.e., don't present your result as more robust than it is; admit that it only comes out this way when you do the analysis a particular way). See the below now-classic paper:

Article False-Positive Psychology

Secondly, keep in mind that .05 is an arbitrary cutoff. You should report exact p-values (at least to a few decimal places) rather than just lumping everything into "significant" and "non-significant". If you look at the exact p-values, your results might not be very different (for example, p=.06 and p=.04 are about the same in terms of the strength of evidence they provide against a hypothesis).

Thirdly, you should think about what is the purpose of your regression model and the variables in it. Often, regression is used to examine how one variable is related to the DV while also controlling for the potentially confounding effects of other variables. If you had an a priori reason to want to control for the effect of the extra variable (narcissistic rivalry), then there is no point in having ever looked at the first model that didn't include that variable.

Fourthly, your results aren't contradictory. It is normal that adding a variable to the model might change the effect of other variables (not just changing the significance; it can even reverse the sign, there are many examples of this online). This is usually the whole point of using regression. Consider the example in the attached image. If you just regress the DV on the first IV (IV1, the x-axis in this graph), you will get an upward-sloping line, suggesting that the DV goes up when the IV goes up. But if you add an additional variable (Group), then you will get a separate downward-sloping line for each group: in reality, the DV goes down when the IV goes up, and it just so happens that there are two obviously different groups but they both follow this pattern. That is a classic sort of case where adding another variable will completely reverse the sign of a different coefficient.

There are a few separate issues here. First of all, you shouldn't try out various analyses and then choose one to report; if you are fishing around with different analyses that weren't planned ahead of time, you should report everything you did, or at least honestly report that the result you are describing depends on certain analysis decisions (i.e., don't present your result as more robust than it is; admit that it only comes out this way when you do the analysis a particular way). See the below now-classic paper:

Article False-Positive Psychology

Secondly, keep in mind that .05 is an arbitrary cutoff. You should report exact p-values (at least to a few decimal places) rather than just lumping everything into "significant" and "non-significant". If you look at the exact p-values, your results might not be very different (for example, p=.06 and p=.04 are about the same in terms of the strength of evidence they provide against a hypothesis).

Thirdly, you should think about what is the purpose of your regression model and the variables in it. Often, regression is used to examine how one variable is related to the DV while also controlling for the potentially confounding effects of other variables. If you had an a priori reason to want to control for the effect of the extra variable (narcissistic rivalry), then there is no point in having ever looked at the first model that didn't include that variable.

Fourthly, your results aren't contradictory. It is normal that adding a variable to the model might change the effect of other variables (not just changing the significance; it can even reverse the sign, there are many examples of this online). This is usually the whole point of using regression. Consider the example in the attached image. If you just regress the DV on the first IV (IV1, the x-axis in this graph), you will get an upward-sloping line, suggesting that the DV goes up when the IV goes up. But if you add an additional variable (Group), then you will get a separate downward-sloping line for each group: in reality, the DV goes down when the IV goes up, and it just so happens that there are two obviously different groups but they both follow this pattern. That is a classic sort of case where adding another variable will completely reverse the sign of a different coefficient.

Stephen Politzer-Ahles wrote:

"Often, regression is used to examine how one variable is related to the DV while also controlling for the potentially confounding effects of other variables. If you had an a priori reason to want to control for the effect of the extra variable (narcissistic rivalry), then there is no point in having ever looked at the first model that didn't include that variable."

I'll just note that in some fields, the approach is to show the crude (unadjusted) effect of the focal variable, then the adjusted effect--and if you don't do this, reviewers may demand it.

Emma Bromley, if the relationship between X1 and Y is strengthened when X2 is added to the model, X2 is often described as a suppressor variable (e.g., in Psychology) or a negative confounder (e.g., in epidemiology). HTH.

I'd also add that in published work the effects of admiration and rivalry are usually tested with both simultaneously in the model to assess their unique effects.