Yes that formula is correct there are different formulas for different d-family measures.

Cohen's d is not bounded by 1 and you could get such effect sizes which according to Cohen is enormous.

In ANCOVA there is also r-family effect size: partial omega squared  used  frequently that would return a value between 0-1.

How many variables/covariates in your ANCOVA table?

Please have a look at links in following discussion and the link

http://oak.ucc.nau.edu/rh232/courses/eps625/handouts/ancova/understanding%20ancova.pdf

https://www.researchgate.net/post/Is_there_any_difference_in_the_effect_size_calculation_Cohens_d_for_within_groups_and_between_groups

Cohen's isn't defined for ANCOVA designs. It is a population statistic that is defined as a difference between two means divided by the population SD (originally the SD of the control group in the population). There are a many formulae used to estimate this quantity (or a related quantity) and the correct formula will depend on what you are trying to do.

Using the mean square error from the ANCOVA seems like a bad idea if you want to get a measure of practical or theoretical importance of an effect. This is because the variance accounted for the by the covariate will be partitioned out from the MSE and thus make it appear larger than in a study without the covariate (or in the raw data). On the other hand, if you want to do a power analysis using the MSE from the ANCOVA is exactly right (assuming you are estimating power for a similar ANCOVA design) as the ability to detect the effect does improve with the covariate (assuming the covariate is correlated with your outcome).

I'd also probably not use the raw means for the conditions (in either case). I would compare the adjusted means from the covariate - otherwise why bother with the ANCOVA?. I think the most natural effect size measure for me would be the differences in adjusted means between the two groups. I would avoid standardising (if possible) as that can easily distort judgements of the magnitude of an effect (because anything that changes the sample SD but not the population SD will bias your estimate of d). If you must standardise I'd probably use the raw SDs of the groups.

See Olejnik & Algina (2000).

Since some people's attention will be drawn to this post in the future I'd think it would be good to say how you can derive effect size using ANCOVA.

In this regard, if you're using SPSS for your ANCOVA analysis, you can check the "Partial Eta Squared" column in the "Tests of Between Subject's Effect" table. This can be used as the cohen's d and must be interpreted in that way (0.8 equals large effect, etc.)

Perhaps late for answer to Ms Grant. SPSS is not giving a direct intepretation from ANCOVA. The best way to deal with size of effect is to calculate eta square (is not the same than partial eta square). The etha square is computed as SS of the treatment of interest (can be a treatment or any condition) divided by SS of the SS Corrected total: SS treatment / SS corrected total.

You can transform the result to f or to Cohen d:

eta^2 = f^2 / ( 1 + f^2 ) , and consider that 2f=d. Therefore, the usual sizes of effect of 0.2, 0.5 and 0.8 correspond to f 0.1, 0.25 and 0.4 and or etha^2 to 0.01, 0.06 and 0.14, respectively.

A site for conversions among different types of size of effects and I found helpful is https://www.psychometrica.de/effect_size.html

Late, but I hope this answer can help in the future.

You could use the Package "compute.es" of the R software, with the "a.fes" function to calculate the effect size from an ANCOVA F.

ANCOVA F-statistic to Effect Size: a.fes

Description

Converts an ANCOVAFto an effect size ofd(mean difference),g(unbiased estimate ofd),r(correlation coefficient),z′(Fisher’sz), and log odds ratio. The variances, confidence intervalsand p-values of these estimates are also computed, along with NNT (number needed to treat), U3(Cohen’sU(3)overlapping proportions of distributions), CLES (Common Language Effect Size)and Cliff’s Delta.

Usage

a.fes(f, n.1, n.2, R, q, level=95, cer = 0.2, dig = 2, verbose = TRUE, id=NULL, data=NULL)

Arguments

f: Fvalue from ANCOVA.

n.1: Treatment group sample size.

n.2: Comparison group sample size.

R: Covariate outcome correlation or multiple correlation.

q: number of covariates.

level: Confidence level. Default is 95%.

You can do the calculation using the function "a.fes" directly on this page:

https://rdrr.io/cran/compute.es/