Logistic regression - odds ratio and significance

I'm working with logistic regressions for the first time, and I'm trying to wrap my head around my results. I've read tutorials and other posts here on CV quite extensively, but I'm still unsure that I have this straight in my head.

I have used the logistic regression to determine how rate of change of a landscape (%) influences the climate-growth relationship (the binary variable - positive or negative) of individual trees.

Here is one of the logistic regression models, the summary, the CI, and the odds ratio that I calculated using exp(coefficient):

> glm_wintermin_neg<-glm(wintermin$Neg~wintermin$thaw, family="binomial")
> summary(glm_wintermin_neg)

Call:
glm(formula = wintermin$Neg ~ wintermin$thaw, family = "binomial")

Deviance Residuals:
Min       1Q   Median       3Q      Max
-0.6391  -0.5989  -0.5967  -0.3720   2.3261

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)    -2.80287    0.55190  -5.079  3.8e-07 ***
wintermin$thaw 0.03136 0.01535 2.044 0.041 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 290.64 on 349 degrees of freedom Residual deviance: 285.28 on 348 degrees of freedom AIC: 289.28 Number of Fisher Scoring iterations: 5 > confint(glm_wintermin_neg) Waiting for profiling to be done... 2.5 % 97.5 % (Intercept) -4.067980576 -1.85254625 wintermin$thaw  0.004366896  0.06591151

> exp(glm_wintermin_neg$coef) (Intercept) wintermin$thaw
0.06063549     1.03185810

The odds ratio I calculated as 1.03, and this is significant (p = 0.041). My concern is that I have other non-significant (p > 0.1) models that have odds ratios that are very close to this odds ratio, but both slightly higher and slightly lower (OR = 1.019, 1.047) which set off some alarm bells. My interpretation is that what my results mean is that the odds of a negative response are 1.04x larger than a non-negative response. The reason for this particular model being significant vs. the others that have similar ORs I believe is because for the significant model, the CI doesn't encompass 1. If someone would be able to confirm my interpretations, and/or clarify areas that I'm off track, I would be very grateful.

Thanks in advance, and if there is any other information, code, or results that I can provide, please let me know.