Odd ratio vs Wald test. Contradiction or missunderstanding?

Question

I have the folowing model:

mod.log.diab <- glm(data = data_train, INFEC ~ DIABETES, family = binomial(link = 'logit'))

This gives the folowing OR:

>exp(mod.log.diab$coefficients)
(Intercept)  DIABETESno 
   2.826087    1.469822 

This value is greater than 1 so we can say that the variable is significant. But when I apply a Wald test I get the folowing answer:

>regTermTest(mod.log.diab, "DIABETES")

Wald test for DIABETES
 in glm(formula = INFEC ~ DIABETES, family = binomial(link = "logit"), 
    data = data_train)
F =  2.34702  on  1  and  1560  df: p= 0.12573 

The pvalue is greater than 0.05 so we reject H0, the test says that is not significant.

How I have to interpret this, who has the truth OR or the test?

The question does not finish yet because if I upgrade the model with more explicative variables:

mod.log.cat <- glm(data = train.cat, INFEC ~ DIABETES + EDAD + HCTO, family = binomial(link = 'logit'))

I get the folowing OR:

>exp(mod.log.mult$coefficients)
(Intercept)  DIABETESno        EDAD        HCTO 
  1.3275634   1.0751666   0.9817025   1.0558760 

And the folowing Wald test output:

Wald test for EDAD
 in glm(formula = INFEC ~ DIABETES + EDAD + HCTO, family = binomial(link = "logit"), 
    data = train.cat)
F =  18.90695  on  1  and  1558  df: p= 1.4615e-05 
Wald test for DIABETES
 in glm(formula = INFEC ~ DIABETES + EDAD + HCTO, family = binomial(link = "logit"), 
    data = train.cat)
F =  0.4122714  on  1  and  1558  df: p= 0.52091 
Wald test for HCTO
 in glm(formula = INFEC ~ DIABETES + EDAD + HCTO, family = binomial(link = "logit"), 
    data = train.cat)
F =  41.87676  on  1  and  1558  df: p= 1.2986e-10 

This time DIABETES does not reject the wald test, like in the simple model. But the strange fact is all odd ratios are near similar, so they are near 1. What is the conclusion? How is possible with OR near to 1 some variabels are significant and others no?

Thanks.