tl;dr: What is the difference between a regression model that allows its dummy variables to be correlated and one that doesn't do so? For example, how does that affect Standard Error of the regression coefficients (coefs)?

In my reproducible SEM (structural equation model) modelA below, imagine I remove the Group_HI ~~ 0*Group_MT (denoting correlation bet. two dummies is 0) altogether leading to modelB. What is the difference between a model that uses Group_HI ~~ 0*Group_MT (modelA) and one that doesn't include anything to describe the relation between Group_HI and Group_MT (modelB)?

Specifically, it seems modelB produces Standard Errors for coefs that are closer to those given by a MANOVA:

summary(lm(cbind(MP,SE)~Group,data=d)) 

Response MP :
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   4.6667     0.6150   7.588 8.19e-10 ***
GroupHI       3.1833     0.8136   3.913 0.000281 ***  ## This SE
GroupMT       6.7451     0.8438   7.994 1.95e-10 ***  ## This SE

Response SE :
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   4.2000     0.3378  12.432  < 2e-16 ***
GroupHI       2.1500     0.4469   4.811 1.48e-05 *** ## This SE
GroupMT       3.7412     0.4635   8.071 1.49e-10 *** ## This SE

But modelA produces Standard Errors for coefs that are closer to those given by:

stacked_DVs <- pivot_longer(d, c(MP,SE), names_to = "DV")
coef(summary(lm(value~DV*Group-1,data = stacked_DVs)))

              Estimate Std. Error   t value     Pr(>|t|)
DVMP          4.666667  0.4961705  9.405370 2.364383e-15
DVSE          4.200000  0.4961705  8.464833 2.559422e-13
GroupHI       3.183333  0.6563718  4.849893 4.648941e-06   ## This SE
GroupMT       6.745098  0.6807403  9.908475 1.912183e-16   ## This SE

Reproducible data and R code:

library(tidyverse); library(fastDummies) ; library(lavaan)

d <- read_csv("https://raw.githubusercontent.com/rnorouzian/v/main/memory.csv")
d <- dummy_cols(d, "Group")[-5]

modelA <- "MP ~ Group_HI
   SE ~  Group_HI
   MP ~  Group_MT
   SE ~  Group_MT
 Group_HI ~~ 0*Group_MT  #Question: what is the effect of removing this line on SE of coefs? (See `modelB`)
   MP ~~ MP
   SE ~~ SE
   MP ~~ SE"

summary(sem(modelA, data = d, meanstructure = TRUE))

Regressions:
                   Estimate  Std.Err  z-value  P(>|z|)
  MP ~                                                
    Group_HI          3.183    0.659    4.830    0.000   ## This SE
  SE ~                                                
    Group_HI          2.150    0.362    5.938    0.000
  MP ~                                                
    Group_MT          6.745    0.684    9.868    0.000   ## This SE
  SE ~                                                
    Group_MT          3.741    0.375    9.963    0.000



modelB <- "MP ~ Group_HI
   SE ~  Group_HI
   MP ~  Group_MT
   SE ~  Group_MT
### Group_HI ~~ 0*Group_MT  #This line removed.
   MP ~~ MP
   SE ~~ SE
   MP ~~ SE"

summary(sem(modelB, data = d, meanstructure = TRUE))

Regressions:
                   Estimate  Std.Err  z-value  P(>|z|)
  MP ~                                                
    Group_HI          3.183    0.790    4.031    0.000  ## This SE
  SE ~                                                
    Group_HI          2.150    0.434    4.956    0.000  ## This SE
  MP ~                                                
    Group_MT          6.745    0.819    8.235    0.000  ## This SE
  SE ~                                                
    Group_MT          3.741    0.450    8.315    0.000  ## This SE